I would bring the green card, esp if I were crossing state lines.

Maybe I have just been unlucky, but I seem to stumble across these "internal immigration checkpoints" more than just rarely. Once on an Amtrak train in Florida, once at an airport TSA checkpoint, and once while driving down a highway in Vermont. In fact, since I am heading up for a July 4th camping trip in the Adirondacks this weekend, you just reminded me to remind my resident PR friend to bring along the resident card. The chances of getting stopped are still small, but I imagine it would so much less of hassle if you had the card.