My idea is that there is no need to try hard fitting normal training into your living room. If you have a whole good room available for that at home, then good for you, but otherwise it doesn't really matter. Instead, homework time can be a great opportunity for doing things that you can't do during lessons. Wait, do I mean there is something you cannot do at the dojo but you can do at home? Not really, but there are certainly things you cannot do when training with the whole class, simply because the teacher and other students cannot wait for you. So unless you are allowed to freely use the dojo space for your own solo training, take advantage of homework time for making detailed investigations and exercises tailored to your personal needs.
#1 - Review lesson material
One idea is to write down notes (or keep in mind) right after each class, about the 2-3 things from today's lesson that left you feeling you need to understand better, or that you simply don't want to forget. Your homework is to go over those few things at your own pace, doing as many repetitions you think you need, without having to adhere to a schedule or switch to the next exercise until you choose so.
Oriental martial arts typically have a very extensive material to learn (techniques, kumite drills, kata etc.), and this unfortunately means there is a good chance that many months can pass before you see a specific technique again in a class, unless it's one of the basics. Reviewing in a short time (same evening or next day) what you've just done at the dojo, and then reviewing it again a few days later, can really help to keep it in mind. Personally I have realized that I have always reviewed kata at home, while I have rarely reviewed self-defense techniques, and the results are nowadays very clear: I have an easy time remembering all kata we've seen in classes, even if some of them we haven't done for a year, while on the other hand I still have troubles remembering the self-defense stuff!
#2 - Slow down everything
Whatever our level and experience, I am pretty sure we are all aware there is room for improving every single technique we can think about. Perhaps when people talk about improvements, they mostly think about becoming faster and more powerful, but it's also quite clear that precision is the enabler of both speed and power. If your technique is sloppy, speed and power will be wasted, and the final result inefficient.
In musical instruments studies, a lot of people will tell you that if you want to get fast, you need to practice slow (not exclusively slow, but still). I think the same applies to karate. Unfortunately, slowing yourself down for your needs is the least likely thing you can do during a class training: the teacher sets the pace, and everyone just follows. If the pace is too fast, there is not so much chance to fix subtle mistakes. Take advantage of being able to set your own pace freely at home, so that you can focus on the details of your techniques, listen to your body feedback, and figure out what you need to adjust.
#3 - Select/adapt exercises for limited space
No reason to worry about not being able to practice a whole kata in your bedroom, when there are so many other things you can do: single techniques or combinations in fixed stance, kihon ido limited to one or two steps in any direction, and even close-quarter self-defense and kumite drills.
But as a matter of fact, even exercises that require more space can be adapted to a smaller area. How about kata for instance? It might seem impossible to practice kata at home, but what if you break down the kata into shorter sequences and practice only one of them at a time? What if you modify the enbusen (steps pattern), changing directions differently so that you never need to move too far? It won't be the same kata, but this doesn't mean it will be less useful to practice. And no, it will definitely not mess up with how you do the full kata in the regular way.
#4 - Physical supplementary exercises
In addition to technique, you can of course also do some straight physical exercises. There are many which don't require to move around at all, such as push-ups, sit-ups and squats. If you don't want to sweat even a bit, stretching is always good.
Also look for physical exercises that still have a lot of karate-feel. For instance, choose a demanding stance such as shikodachi or nekoashidachi, and try to stay there for a few minutes in a row (pausing when necessary). This is actually a good alternative for strengthening the leg muscles, and it doesn't even require more space than the one for standing up!