My opinion - You need good maps! You need a good map of the cave and also a good map of the properties over it.
A map of properties should be available at your county's web site. Most counties now have all of their plat maps, property parcels and related information on a GIS system.
A map of the cave could be more of a problem. I think the place to start is your local caving group. You can find them at the NSS web site
https://caves.org. There may be a state speleological survey group which maintains cave information. The local caving group will know about any state survey group. There may not be a good map of the cave, in which case you have a really interesting problem!
With these two maps, you should be able to overlay one on the other to see what of the cave really is under your property. From there ... Lawyers and $$$.
Bill Gee