If you need your ships to repair ~200 hp in a very small amount of turns, you need better ships-which may mean that you need better techs. They shouldn't be taking that much of a beating-or rather, they shouldn't be taking that much of a beating and still surviving (for instance, about the only scenario I can see where they might take that much damage would be against DL ships, in the campaign or something, which tends to destroy them outright).
But there is nothing else that you can do, short of modding, to increase your repair rate, aside from what has already been said.
I do seem to remember them repairing when they go up in level, but that could just be the extra hp that they gain, and not full hp. Apologies if I'm misremembering.