I would float lightweight metal track & stud framed rooms right off the ceiling/joists. Basically, build whatever sized box, lift it up, and screw it in. You can't just go framing a basement out with lumber afaik, or it could lift the house off the foundation walls. Anyway, Insulate it all up with rigid foam and install some kind of thinner paneling inside and out of whatever size room/partitions you make. . FRP/plastex on the inside if possible, maybe even glued right over the foam with a compatible adhesive to shave weight. Designed to be floating, as in it hangs an inch or so completely up off the floor, and also so attached in such a way you could take it down and install somewhere else someday if needed.. Will never rot out. Leave air gaps between the new room\s and any existing walls so it can still breath around it, and still allow the concrete to permeate. Then, just stuff or spray the small gaps down below/between the bottom track of the metal framed walls & concrete floor with more foam, so it can still expand and contract up and down.. like some basements do. Or, even frame a floating floor right into the hanging walls too, with the same studs going horizontal like joists (they do it in metal framed tiny homes for lofts/bed areas/etc, even though its cheap non load bearing drywall studs & should hold whatever sized grow up np), so your up off the floor a few more inches. Or, just slide a whole foam panel under, and put laminate flooring on top only in that section..
Something about trying to seal up a damp basement from the inside with whatever products doesn't sit well with me, but then again I never had one. Seems like it would cause more damage and cracks. Seems you would want to water proof it from the outside walls if anything, which involves excavating all around the house first. Right? Do you guys have radon gas in the ground in your areas, or need it to be mitigated? Basements are creepy to me. I wish i had one though, lol. Or a bunker..
