I am a little concerned that pumping too much of the foam through a small hole might cause the drywall to bulge.
Not a problem. The way that stuff works it comes out goopy and bubbly and expands. It doesn't push, just spreads out in all directions, path of least resistance kind of thing. You'd have to pump a whole can in there to have a problem. Maybe not even then.
The more likely problem is it goes in and by gravity or whatever just doesn't fill in the right spot. But you're really only out the cost of a can so what's to lose? Worth a shot.