This is why we recommend interior solid core slabs in a "flush" configuration. That is, no divided panels at all. It's a function vs. fashion deal here.
While your doors are made of Oak, and intuitively that's pretty solid, however as you see they aren't.
OK Solution: On the theater side door surface, add a sheet of plywood, MDF or high density (underlayment grade) particleboard. You could consider some Green Glue there, but there's limited surface area for contact. You will also be trapping a small amount of air, which will be problematic as well.
Good Solution: On the same theater side door surface, completely fill in the recessed panels with automotive Bondo or epoxy. Then GG a panel to this finished surface.
Better Solution: Green Glue the recessed areas first, let GG dry, then pour a leveling epoxy over recessed areas to create a single flat surface. Then GG a panel to this new surface.
Best Solution: Replace the door slabs with solid core interior slabs. These could be particleboard, MDF or mineral core slabs. In actuality, this isn't as sound isolating as the "better" solution above, it has the highest performance to PIA factor.
The Soundproofing Company