I'd lean towards a fabric hinge too. I'd suggest using two strips of fabric seamed down the middle, so that you can glue one flap to both angles of each side of the mating
line (think of the fabric strips in cross section as an X, with the upper jaw in the top quadrant of the X, and the lower jaw in the lower quad), so that the fabric has less chance of starting to peel and pulling free at the hinge
line, plus it doubles your contact area. Make sure to use water-based glues, anything
solvent based will just attack the foam and make matters worse. If the mating surfaces aren't
flat, cut the fabric strips on the
bias (diagonal), and then stretch the fabric to match the curves of the foam.
If you want to try mechanically anchoring the fabric as well as gluing it, drywall screws work disturbingly well in extruded foam. Numerous 3" screws and fender washers could help stabilize things.
I'd be careful with trying to use spray foam to
build back missing foam. Some brands are
solvent based, and all of them tend to keep expanding for a while after spraying, and could do as much harm as good if you're trying to fill a confined space.
As for glue, you'll have more problems getting the glue to
grip the foam than the fabric. I'd look for something that's somewhat flexible when dry, rather than a brittle white or yellow glue. Tacky Glue comes to mind, as does water-based contact cement, or even just using unthinned latex paint.
I'm assuming your theater owns the puppet, since you're talking about repairing it yourself. Many theaters rent a set of Audreys, and any repairs or modifications would tend to make the rental
house upset.