How to re-install?

You will have to add wood into the hole and drill into that for additional support

If you can place a couple of wood strips inside the opening and attach them to the drywall then attach the panel to the wood strips with screws.

Liquid nail for tub surrounds

I have used these panels many times and have installed them using a latex caulk. As long as the hole is small enough to make contact with the back of the flange and there is sufficiant caulk contacting both the ceiling and the flange it should stay in place. You may have to support the panel in place untill the caulk sets up. Using latex caulk allows you to clean any excess with a damp rag without any residue on the ceiling as long as the ceiling is painted with a paint that will be able to be cleaned. If the hole is still too big, you could get a larger panel and cut the hole to fit that panel