If the underlying cause of the behavior is a lack of food (due to it being denied), is there a practical way to resolve it with rather limited manpower?

There are two ways to solve misbehavior: fix the underlying problem that causes the misbehavior, or tell (force) the cat to stop behaving this way.

The former is much preferred. Since its behavior is usually a consequence of the environment it finds itself in, if you don't fix the environment but do force the cat to stop their (in their mind justified) response, then they'll see you as the bad guy.

However, kittens are impressionable. If they've grown up in a hostile environment, even when you remove the problematic situation, they may stick to doing what they know because they've never known anything else. In that case, you will have to retrain them even after fixing the underlying problem.

If the underlying cause of the behavior is not a lack of food, is there a way for it to be corrected?

Whatever the cause of the misbehavior is (food seems a good first bet but it could be something else), you should first fix the issue so the cats no longer need to behave this way in order to survive. Once you've done that, if their behavior persists out of sheer habit, it becomes time to retrain them.

The only way to do so is to use negative feedback, just like how you would with a dog. Cats will eventually learn from your feedback but it takes much longer than for dogs because cats don't have the innate instinct to listen to their owner.

There are many ways to do this, I tend to apply these responses based on what's best in context:

• Raise your voice. Call them by their name and follow up with a clear no.
• Take away the food. Either give it back after a short while (it's enough to send a message), or give it to the cat that was being victimized (which sends a clear message that you will favor the victim over the aggressor - crime does not pay)
• When the cats get physical, get physical back. Do not hurt your cats. The goal is to send a message. A soft push to bring them off balance is more than enough to send the message. I tend to push their butt to the side as they are less able to dodge that.
• Get a plant sprayer and take off the cap so it squirts streams of water. It's a very effective ranged tool to get cats to stop doing what they're doing.

Some sidenotes about this:

• The general approach in all of the rules is to inconvenience the cat rather than full on punish them. When you punish them, they are liable to see you as the aggressor, and then you're just adding to the hostile environment.
• When two cats engage each other in a fight, both should be punished, regardless of who started it. When one cat is only defending itself from the aggressor, only the aggressor should be punished.
• For long term training, always start with a verbal warning, and only resort to physical/water responses when that verbal warning is ignored. Over time, they will learn that ignoring the verbal warning means a stronger response is coming, and they will back of when they hear a verbal warning.
• Remain consistent. When you've decided to punish a certian behavior, always forbd it for every cat. This sends a clear message of zero tolerance.
• Pick your fights. If your cats have many misbehaviors, don't address them all at once. Pick 3-4 topic and prohibit those. When they make progress, add remaning topics. If you punish everything all at once, the cats might end up seeing you as getting upset about literally everything and your message will not reach them.

It is not practical for us to feed all of the cats separately.

While I can understand that, there may be a lot of benefit gained from sometimes splitting them off. For example, if one cat becomes needlessly aggressive, putting them outside while the others are being fed inside sends a powerful message.

Don't starve your cat, give them some food in a short while or let them back in so they can still have some of it. But that short period of missing out on food right after being violent with others sends a clear message that violence does not pay.

In many ways, you can teach a cat the same way you teach a child that chooses to not listen to what you say. Since cats don't speak your language, they should be treated just like children who refuse to listen: learning through practical experience.

Is it possible / likely that this behavior is the product of a couple generations of inbreeding?

I can't say it's impossible but I do consider this highly unlikely. Aggressive behavior isn't a genetic defect.

Or is it more likely that this behavior is due to them being denied access to food by the other cats when we aren't looking?

This is much, much more likely.

I have friends and family that are interested in taking care of these two kittens.

If friends and family are willing to train the cats, it may be easier to train them when they're young and in a new environment. They will already have to adapt to the new environment, might as well make them adapt to new house rules (no violence etc.) at the same time.

This may also solve an issue of cat overpopulation. Cats are territorial, and having 15 cats in the same place is likely adding to the social friction between the cats. They don't have their own domain and that's going to cause issues such as not putting up with each other.

If the behavior persists as they grow up they will become a genuine threat to safety.

Barring cases of cats that were egregiously abused by their owners, I have yet to come across a cat whose behavior cannot be adjusted by proper training.

"Spare the rod, spoil the child" is a good general approach when used figuratively and not as a justification for corporal punishment. Never EVER resort to physical abuse.

I've used what can be considered classic child raising methods for cats with great success. This includes the punishments I've mentioned before, but also:

• Escalating punishments. When you're implementing a rule that didn't exist before, start with soft warnings. As the cat keeps breaking the same rule over and over, the punishments escalate as they "should know better". But it goes without saying that you need to remember that cats do not speak English (or your language) and non-verbal communication can take much longer to pass the message along.
• Repetition and consistency is key. Never allow something because you can't be bothered to punish it. Don't send mixed signals. Make sure all humans in the house are aligned on what's allowed and what's not. For example, if your partner lets the cats on the couch but you don't, the cats are going to end up confused and not getting a clear message.
• Timeout room. Cat misbehaves, goes in the timeout room. For warnings, I take them to the timeout room but do not close the door. Merely being carried to that room is enough to get them to realize they need to stop whatever it is that they were doing.
• Understand that your cat is not trying to refuse your authority - it simply doesn't comprehend the idea that you wield authority over them. Cats tend to act on their own decisions, and thus will always be genuine. Never punish a cat without first trying to understand why they're acting the way they are (but physically assaulting others is something that you can respond to immediately, of course).

There is a lot more to be said on the topic of cat training, more than I can put in a single answer. I suggest you browse the cat questions on this site as this has been addressed before in the past.

This is an answer I gave a while ago which goes more in-depth as to how you train a cat through negative feedback. Chapter 4 is the most relevant for your case.

As an addition to Flater's answer:

It is not practical for us to feed all of the cats separately.

Feeding many animals from one bowl or in one tight spot creates stress because the weakest are forced to wait for the strongest to finish before they can eat.

You should put out many small bowls of food and spread them over a larger area so even the weakest have the chance to eat while the strongest are still eating. This reduces stress and aggression.

The same principle is used in many zoos today to prevent stress and additional work for zookeepers. In the past, they had to separate individuals during feeding time, now they can feed the whole group at once.

I want to add something to the excellent answer given by @Flater

It seems that inbreeding may actually result in aggressive behavior in cats.

I was referring to What are the disadvantages of inbreeding cats?

As mentioned by Jennie Balfour about his/her cat Tigger

He’ll snuggle up to them & should they wake & try to wash him, he’ll tolerate it for a few seconds & then attack them.

He’ll eat any type of cat food that’s put in front of him & if there’s not enough on his plate, he’ll attack the other 2 cats, drive them away from their food, & quickly gobble it up.

To this day, he’ll target any cat that dares to cross the property line & fight them.

You can’t pet him for too long since it overstimulates him & he’ll bite.

He also has a very different personality from any other cat I’ve owned (& I’ve been a cat owner my whole life). He’s more animal-centric than he is human-centric. How much of that is due to coming from a feral colony & how much is due to being inbred, I can’t say for sure.

Here I can see many similarities between the behavior of Tigger and that of your two kittens.

In the site PROBLEMS WITH INBREEDING CATS

It is mentioned that inbreeding may result in mental problems for cats.

The Cat Fanciers Association of America notes that inbreeding may lead to immune deficiencies, more congenital abnormalities and cats that don't grow to their potential.

The site Mental Retardation in Cats also mentions that inbreeding may result in mental disease.

The site Retardation in Cats is a cause of mental illness in cats.

All cats can inherit diseases if both parents have the same gene mutation, but the likelihood is greater among pedigree cats bred from a restricted gene pool

As a SIDENOTE:

You mentioned in your question that

One of them recently slashed the other's eye to the point where fluid started oozing out after it tried eating with it.

So regular trimming of their claws is definitely advisable.