How to make ice cream base using evaporated milk easier to work with?

When creating ice cream you always have to balance a couple of parameters of the product. The two of most importance of them are the desired sweetness and the freezing point depression that directly affects the hardness at the serving temperature. This is usually gained by choosing a mixture of different sugars that differ in their sweetness and anti-freezing properties. As I nowhere read "sugar" in the ingredients you list, I assume you just rely on the sugars already contained in the other ingredients, which are some sucrose from the choclate and some lactose from the dairy. They are both disaccharides and both have a medium freezing point depression. In general I would recommend to replace some of the sugar in your mixture with dextrose/d-glucose, that lowers the freezing point much more effectively than sucrose. Adding more milk powder also would have a similar effect, as it is high in lactose content that has the same anti-freezing effect as sucrose but a much lower sweetness, but you should also aim to not exceed the amount of it over ~10% of the total mixture as it then could affect the texture in a negative way.

If nothing of this is an option adding some alcohol, ideally some sort supporting the taste, also could serve this purpose.

I don't think you can use this recipe with a standard machine.

Ice cream machines are supposed to only freeze ice cream to a certain temperature (the "draw" temperature) and the rest happens in the freezer. To ensure that you don't overchurn, modern cheap home machines have some kind of sensor for the resistance of the mass, and professional machines have a special type of bearing such that the dasher stops moving at a certain viscosity even though the motor continues turning. Machines in between those two classes (such as home compressor machines in the 200-2000 Euro range) are notorious for having short lifetimes if not stopped before the ice cream has frozen too hard.

I am pretty sure that the sticky mass you are describing has higher viscosity than your machine has been designed for, so you won't be able to continue with this machine.

You could try a manual method (either entirely with your hands or with a hand cranked machine) or go some kind of DIY route and build something to churn for you, using some kind of motor adapted to the task (high torque, low speed). This may not be sufficient though, since a normal dasher design won't be able to exchange the just-frozen mass from the wall with the warm mass in the middle when the viscosity is too high. In the end, you might be looking at either direct freezing (which will still produce something scoopable with this kind of recipe, just not too similar to what we usually regard as ice cream) or high-tech tomfoolery with liquid nitrogen and the like.

This is very anecdotal, but I finally finished fiddling with this batch. Throwing the mixture into a blender made it flow a lot easier - wildly guessing long protein chains got broken up, it didn’t gain much in volume for aeration to explain this. (I might try whipping up in a future experiment but my stand mixer is close to falling apart.)Still very thick, but less paste-like.

It also churned without messing up the machine for 40-50 minutes; but moving between containers made me lose a significant amount of the base so that could have contributed.

As far as the final texture goes: marginal improvement if any. Still not easily scoopable, and very “chewy.” (To each his own, but my teeth are acting up this week and really don’t like biting into anything cold.)