Multiple "brain-backups" in the spine, how would it work?

But how could such a redundant spine work and be good at its job (keeping the dragon's memories intact and heart beating)?

Redundancy. The dragon has effectively several "brains" along its spine, and they coordinate much like a human's hemispheres, only better. The spine is actually two spines - the main motoneural/somatic spine and a secondary bus similar to a human's corpus callosum, through which the brains synchronize. In any moment, the "dominant" brain controls the main spine.

This would cause all sorts of interesting side effects, like splitting a dragon in two resulting (after a lengthy recovery) in two identical dragons with the same memories.

(And yes, it would not work with actual nerve fibers, because neurobiology. And makes no evolutionary sense. But it sounds nice).

Have you considered compromising a bit on what is backed up? If the aim is to make it so that the dragon can survive an instance of going berserk, is memory really that important?

Let's scrap the "memory backup" side of things for a second and consider the other options. If all the backups have to do is keep the body alive (and potentially moving in some basic manner), you could have a secondary nerve cluster that controls "life support" - you could even have it so that this is always on, and that the "main brain" just does deliberate movement - things like lungs and heartbeat run on a completely separate system unless overridden by the brain.

This would give your dragons the ability to survive such a critical brain loss. There is then the compromise to consider. The majority of animals are born with an innate understanding of how to navigate their environment - foals can walk within hours of being born, and there any many organisms that can both move and eat without a recognised brain. That said, most predators I'm aware of have to "learn" to hunt through play, and watching adults - you would have to consider whether to ignore this, or have the dragons need to be retrained every time they die (either magically or it being a lengthy natural process).

This could introduce an element of risk to the whole situation for the dragons - it's available to them, but it's a last resort. Who wants to lose all memory of who they are? If you want to you could include a magical backup of their memories held in a safe store somewhere to be reloaded in event of brain loss (maybe the "life support" backup system has a built in "phone home" impulse which impels them to return to where there memories are?), but the idea of there being a danger to what they would have to do would help balance it out as a mechanic, otherwise why wouldn't they just use it all the time?

if i were you what i would was make it a bit bulkier with a super thick spine that's hollow with a brain cell stem at the top and then it duplicates its brain piece for piece every-time its injured and drops it down to the bottom of the spine and also give it accelerated healing around the spine base to protect the spare brains and then if its brain is damaged it can rely on that for the time being while regrowing its brain up top

If I were genetically engineering a fighting dragon and if I were good at my job, I would segment the brain multiple sections, keeping all of the long term memory in the chest behind the best armor. I would then mount smaller centers of consciousness near each of the major sense organs so that each sense had the total attention of a dedicated brain. Those smaller brains would live in the moment but have smaller sense-specific memory storage so that sensory input could be recognized against past history before being reported to the central, master cognitive brain which is also located in the well defended chest.

This modular approach would also allow for more sensory modules to be hosted by a single creature. Instead of two eyes in the head, mount a pair on the front of the shoulders and a pair at the end of each wing. The tail could have its own cluster of sensory organs for watching/listening the enemy during retreat. You could even keep a pair of eyes in the head for normalcy sake, but except for those eyes, the head should be entirely dedicated to combat with sharp teeth, thick scales and spines.

As a final adaptive survival mechanism, and this is the part which answers your question about brain-backups, let each dragon lay an egg containing all of the material needed to completely clone themselves along with all of their accumulated life learning and memories. You have stated that your dragons are products of genetic engineering and that they have nearly non-existent population growth. Why not explain that racial weakness by repurposing their reproductive systems towards self-preservation rather than population growth. It is much easier for a sentient being to charge into battle and go grizzly if they know that they will quickly be reborn in a new healthy body should their current incarnation be lost in combat.

...and if they survive the combat and therefore don't need to be cloned, they can always consume the unused egg, resupplying their bodies with the necessary components to lay another egg before the next battle begins.

If the dragon isn't based on typical Earth biology of vertebrates, then it might be more like a nematode worm. Worms that are taught to navigate a maze (or some other simple task they're capable of), can survive extensive brain damage. For instance, if split into two halves, when those regrow into two new worms, each will remember how to perform the task.

If such a trait could be put into a larger, vertebrate animal, basically it would only need to survive that damage until it could regenerate, and it would have all its memories/training/personality intact.

The real trouble is that for bilateral animals, all the sensory organs and eating mechanisms cluster around the brain. So it's not just brain damage either. Damage that extensive leaves the animal blind, deaf, and probably unable to eat. So I think you've got the neurological/regenerative elements covered, what you really need to do (should you want realism) is explain how it survives for what will probably be weeks, long enough for those organs to regenerate so it can begin feeding again.

Given that your dragons are sapients with technology (and a competent military hierarchy), maybe they're just scooped up off the battlefield and put on IV drips for 6-12 weeks though.

If you were writing some in-depth hard science fiction, you'd probably want more details, but this sounds like RPG stuff. The details would probably deal with how to have the backup brains receive all the sensory input for memory... biological nerves make for slow data transfer.