ModuTile is a good option. I'm still in decision hell myself. I put down eco rubber backed carpet tiles. Good quality but a year later i had a small flood. So they all went out on the lawn to dry, it rained, ...lol. Back down and another small flood. (just the back 1/5th of the floor) but still enough to be a pain. My moisture issues are almost solved in that area but i still have some research to do. An exterior corner that is a few feet below grade needs to be excavated and dealt with properly from the outside. New hot water system and washer hoses solved the earlier issues. Still, a snap system floor meant for basements is the way to go. Getting a good bond to the existing slab might be a loosing battle. When i pulled up the original owners int/ext wall to wall, The grid lines and glue are visible from the original vinyl tile that obviously failed. I've considered the rubber snap together for the back half where we have storage and workroom. I don't want anything that would absorb moisture. Here is a link that might be useful: rubber tiles

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.

Another here who has had basements all my life and never had a wet, humid basement needing a dehumidifier. Even when I lived on a flood plain. I essentially now live in a Canadian desert (Southern Alberta) and a dehumidifier would likely mummify us. We have a humidifier built into our HVAC to help add humidity especially in winter. LVP is a good option if you have a well insulated basement/slab. If this is a new build it is more likely to be with updated codes, etc but check. I have some friends who have LVP in basement of their new build and if looks lovely! I personally prefer carpet for basements, but that is just a thing for me with private spaces - they get cozy carpet and more public living spaces get wood, laminate, LVP, tile, etc. But I generally prefer to be barefoot so factor that in. You won’t find me putting on slippers in my own home as that is opposite of comfortable for me. When we finish our basement it will be with carpet but it is going to be a private “man cave” space/not a walk out with high traffic. This may just be a “you know what you know” thing as I have always had basements with carpet and to me it just reminds me of coziness, privacy, feeling wrapped in a blanket, etc. A bonus about LVP is if you DO have flooding it can technically be removed and dried out and reused (assuming “clean” overland water and not a sewer backup). Carpet is going to need to be torn out and replaced. But, eh, if I get flooding I’d prefer just to start over.

If you have a flood risk in the basement, I'd say something waterproof like tile or terrazzo. For the playroom area, put down a big cheap area rug (like a section of basic carpet that's not nailed down). Then if it floods again or the kids make a gross mess or spill, you can haul the rug out and get it cleaned, and the floor underneath will be unaffected.