I was really thinking about getting a 4-bike carrier that attaches to a hitch in the rear of my Patriot.

Does anyone know if there is enough room to open the rear door fully when that bike carrier is just tilted down, which most of them do?

Wish I could help... I just went with the roof rack option primarily for the reason you stated. I would find it annoying to try to get around the bike rack, whether it folds down or not... plus your bikes get filthy if you drive on a dirt road.

Good luck though! :D