The guitar is a Hagstrom from 1964, so I'd bet the pickguard is plastic. Hagstrom were great users of plastics in their electric guitars and accordions, and the "mother of pearl" inlays on the acoustics are mostly plastic perloid too.

Stacking books on them does nothing for vinyl records that have warped (I've tried it) not even with cautious warming, so I would not be optimistic. I thing the only way is going to be a pretty determined glue and clamps, and to make matters more tricky the glue needs to dampen the soundboard as little as possible.