I recall a model maker neighbor used laquer thinner to bond plastics,
but that's been fifty years or so ago.

I would suggest visiting a big hobby shop. I was in one recently and
saw a rack filled with specialized adhesives and a counter manned by
hobbyists!

As to the wooden base, might you rout a groove to take the plastic?
Recessing the box into the base might serve to hide the joint and the
glue used.