I do have hyperhidrosis, and like you, I don't think its a main factor in my Body Odor . I can be completely dry and still reek... but I guess it definitely doesn't help when I get sweaty :P
There are a lot of peroxide horror stories online! Benzoyl peroxide is Safe though if used responsibly (a really good derm told me to keep it on the face for a few minutes depending on the concentration, the chest and shoulder area for 10 minutes, and the feet and palms for half an hour).
This surprised me for sure, but he seemed to know what he was talking about and he has some pretty impressive reviews online.
The BP has removed skin odor and slightly decreased hyperhidrosis as a plus. It's been great for me so far, but who knows what the future will hold, and everybodys different, so maybe it won't help other people :/