It seems like you’re simply training too heavy. You can’t do your 5 rep max for several sets, otherwise it wouldn’t be a max. But if you’re creeping into a territory where you don’t even have 1 or 2 reps in reserve, and you’re doing that for several sets across, it seems like your muscles might start failing on the next sets. Also, 10 minutes seems a bit impractical as a rest period. At some point you will need to accumulate volume to improve, and you can’t do that if you’re working so close to failure that you can’t even do the work.
What’s your program like? It seems to me like you simply need to work in a more productive zone percentage wise that allows you to complete the necessary volume in a sensible amount of time. Conditioning and work capacity will improve with time as well, but if you’re working so close to failure, it will be difficult to develop them, as you will simply be too exhausted to complete the session.