I prefer sintra or FRP, and you can use masonite.  The problem with so many plastics (like DVD covers) is they don't hold up long term.  They get brittle and crack/break.  I cannot tell you how many older Puppet Production puppets I have had people bring me and ask if I can "fix" the mouth.  Invariably the thin plastic has cracked, leaving a weird look/feel (and this is glued to fabric).

I guess if you are making the puppet strictly for your use and you know what kinds of temperatures, etc., it will be subject to, it might be OK.  Many groups (teams, church groups, etc.) leave their puppets stored in rooms that are not climate controlled all the time and this takes its toll on the plastics.

Shawn, in Australia we would call that Masonite, it can warp over time if humid but fairly rigid and light. I'm right with you on Balsa, it will snap along the grain. You could try laminating a couple of layers with the grain across each other but it would still be delicate.
I too have tried ice cream container lid in the past and found it twisted when the fabric put too much pressure on the palate. I've been using foam core and am fairly happy, light wait and rigid enough but I'm keenly aware that if it breaks the whole puppet is ruined if you can't get the mouth plate out. If (and it's a big if) it were ever used professionaly it needs to be maintainable and also foamcore is laminated with paper, If this breaks down with sweaty hands the plate is only as strong as it's weakest link.
A friend of mine worked on the original professional tour of Avenue Q in Australia. Their puppets (made in the original US workshop had thin plywood mouthplates with tiny drilled holes to sew through. I'm nearly ready to try some 2.5mm plywood (but I'm not going to drill the holes, glue will do for me.) The same friend has gone on to start a business called Avenue Q hire Australia (find them on facebook) he is making sets of the cast for hire to amateur productions but they have to be robust enough to let out of his sight and expect them back in working order. He is making the mouth plates out of perspex (acrylic) but still hinged with duct tape!