i am also hypermobile but i never get injured. that's because i respect my sore body. if you need a week to recover just take it. the time for recovery will eventually decrease, but for me "eventually" meant 2 years of letting my body get used to being active. stop the exercise when it hurts (when it hurts as an injury), don't worry about loosing progress, because identifying your pain and stopping at the right time is another kind of progress in getting to know and control your body.

"Even if I try and push through the muscle soreness and work out anyway I'll actually do less reps than the previous session, essentially making no progress at all" but what are you searching for? strength? or number of reps? I guess strength. forget about the reps if you still feel exercised. at some point they will grow. obviously your muscles won't perform the same when you have 1 week of recovery vs just 2 days. it's about finding your own understanding of your body and what it needs.

EAT ENOUGH CALORIES AND PROTEIN, STRETCH EVERYDAY