Requesting a recommendation before the end of the course
I don't see how what you suggest isn't worth a shot, if you explain the professor your reasons? It's what I would naturally try. I would go further and offer to hand it in early. Just be ready to back off gracefully if you hear that he is too busy now or doesn't want to do it like this, in which case I would ask if he has suggestions how to handle this respecting both your schedules and commitments (after confirming that he is open to help in principle).