Should I choose someone as one of my references while I also have applied to his group?
Since you seem to know him pretty well, you might think about just asking him. He knows you need a position and that he might not be able to give you one. Yet he supports you, I think. You also would prefer to work with him. As long as all of that is clear and in the open and can be stated, it seems fine to ask for a recommendation. 

But that is the sort of conversation to have face to face, I think. It might not be possible to do that as he is busy, of course, so you might have to resort to email. But if you do, lay it all out so that he understands you are applying for a "backup" situation in case he can't hire you. 

But if any part of my analysis is faulty, you might need to be more cautious and omit him as a reference for now. You can ask again later if nothing current bears fruit.