self leveling layer with internal French drain
How long do you plan to keep the house?

I'd just get the tiles tested and remediated.  How many square feet are you thinking about?

Self level isn't cheap and if you want to make your basement usable at some time in the future you probably want to do something more through.

I've fixed up a number of 1920 basements that had sump pits for ground water collection, no vapor barrier, brick / thin concrete or some other kind of weak terrible floor.

Generally I'd remove the existing floor, remove some dirt, add interior perimeter drains, add a modern sump pit, add a drainage layer, add some insulation, add a radon/vapor barrier, add a pipe for future passive/active radon mitigation, re-pour slab.  Sometimes I underpin the foundation to gain height.