The problem with keeping it permanently damp will be the growth of mould. I don't have any expert advice to offer on preserving this unfortunately, but I do know that mould will usually quickly grow on a damp and dirty surface (and unless it's never been played with, the foam will have lots of dirt / fungal spores on it
Hopefully someone will be able to offer you some more expert advice, but if they don't, I have just found this website which might be helpful:https://www.therpf.com/showthread.php?t=104745
It's an old discussion thread, and they are discussing preserving foam latex. I have no idea what exactly the foam that came with the Dagobah playset is made of (it might be foam latex, but I'm guessing probably not?), but there might be some good advice to be found above nonetheless.