To me NOT sweating (as in: never sweating, a disorder disrupting the horses ability to sweat) is much more concerning than sweating too much just because sweating is basically your bodies' cooling system. Think about things that burst into flames because they get too hot and can't cool themselves, well I doubt pony is going to turn into a fireball BUT it does increase the risk of health issues DRAMATICALLY and can be seriously detrimental. But that's where my 'vast knowledge' ends
. In terms of your lesson horse... well maybe it's just too hot? Unless that horse also has an issue with sweating and is predisposed to sweat too much.
Now to the OP: it is physically impossible for your horse to become 'flabby' after a week off, just like it takes time to build muscle, it takes just as much time to lose muscle and build fat. Maybe YOU believe your horse looks a bit 'flabby' because you're convinced you need to be on a strict routine, but I agree with the others that say your horse needs a BREAK. One day off wont be detrimental to your horse, in fact a day off to mend will HELP your horse build muscle and stamina. Muscles build by ripping and healing, he needs time to HEAL and re cooperate. On top of that, I strongly dislike jumping my horse 2'9 more than once a week, I feel it is detrimental for his joints and I want a competitive, sound horse well into his teen years. By over jumping him you can be causing all sorts of soreness, potential arthritis and lameness.
It sounds to me like he's burnt out, he cannot handle everything you're dumping on him, and by continuing with the 'super strict' regime especially in a hot area you're not going to boost his stamina, just sour him.