Sorry to hear about the damage. I wish I could replace it, but it's unlikely I'll get to use that X-ray machine again. We have hi-res digital copies and a few ideas to mull over for them, but once these originals are gone, they're gone.
The base film stock is a heavy plastic or acetate. I don't know the exact material, but it's basically thin sheet plastic of some kind. Pressing it between two sheets of something is probably a good idea.
I'd suggest something like maybe low heat (hair dryer or heat gun on low) or to leave the pressed sandwich somewhere warm (attic on a hot day, back of an SUV on a hot day, etc.) but really, I have no idea how that'd affect either the base plastic or the image itself.
I may be able to get some "scrap" X-rays from the same source (as in actual pics of bones and whatnot) and do a little experimenting, but no promises.
Two sheets of clean glass, some good weight and a couple of warm-to-hot days might be your best bet.