If the tiles loose, you cannot place anything over them. Question is "how loose" are they? And...will they stay well bonded?

Good question - some are entirely loose, but the rest? Not sure how well they will stay down. Ugh, it's always something, isn't it? What happens if we put a leveling compound over the top of them. Will that keep them stuck down?

No.

Any suggestions? Do we need to put a subfloor over the tiles? What would be a good choice there that is as thin as possible. In a basement, every inch counts.

Take out the tiles, prep the substrate and go from there. As for applying a self-leveling underlayment over the existing tile...you'd have to examine the bond of the tile to the slab to determine if it is likely that the tile will remain bonded. If most all of it is welded to the slab, then proper prep and priming of the tile would be necessary before applying the leveling product. I believe most cementious underlayment manufacturers frown on applying over tile, but that may have changed. Ask them. Their technical departments are good at giving out good information.

It's really a matter of how well the adhered tiles are stuck. If you can get something like one of those roofing tear up bars underneath and pop the tiles up in full pieces, you aren't likely to generate much asbestos dust. The adhesive can have asbestos also so it could be wet scraped and picked up with a wet vac with a capture bag.

If the tiles are adhered such that they break into little pieces, you can likely float where the loose tiles came up and put a new floor over the top.

There's also floating vinyl floors that sound like an ideal solution for your situation but I know little about the efficacy or quality of these products.