just catching your thread...
i would say pond liner or rather epdm roofing liner (for flat roofs) it can be had fairly cheap there is different styles of rubber/plastic out there some cheaper than others pvc, for example.. lay it down, and cut it / fold it so it comes up the walls a 1" or so. maybe a bit more until ya get your walls in. and then put down a floating tong and grove flooring. no need for any sort of shims or like, it goes down and lays right to the curvature of the concrete. and that is how it is. no need to put nails / bolts / screws down into the concrete and through the liner.
any sort of "paint on liners / water proofing" stuff for concrete *big frown* prep work prep work prep work. and again prep work. if ya goof any place, it will peel, bubble away. since your water / moisture is coming from the bottom and up through the concrete... any sort of hydraulic pressure / water pressure, is going to be push up on the paint on stuff. and i just do not see the stuff holding up long term.
6mil plastic :/ does not last, to easy with even slightest of vibrations to tear it / put a hole in it.
if you want to use tile, then just act like it was a customized DIY shower / tub. and use a little bit of tile mud, and set your tiles down into it. and forget the plastic/rubber being placed down.
if ya going to install radiant heating, easiest way IMHO (in my opionin) get yourself some hardboard insulation say 3/4" to 1" thick. using 1/2" pex piping. lay the insulation down on the floor if ya need to score it across the top/bottom so it lays a bit tighter to the floor, use some metal/foil insulation tape over the seams. and then take a router. and grove the hard board insulation out, for were you want to put the pex piping. then place your floating floor (tile or tongue and grove floor on top).
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but before any of above...i might encourage, cutting up the floor. and installing a french drain like doing, around the perimeter of the basement (along the outside walls) and running to the sump. it is costly, but it will drain water from the soil below the floor. this is rather costly doing, but ya need to deal with that moisture issue first. give it a few years, and that moisture = mold. just trying to cover it up, may never work.
double checking.... the 2 sheds here. will sweet. more of a high temperatures / cool temperatures of morning / night time. toss a fan down and it goes away ((more of water condensing on the floor, more than anything)). certain areas are worse than others. due to how the air movement goes through the sheds (pole barns not heated or cooled). it does no real good running a de-humdifier, if the air current is dead. in multi areas of the basement. central A/C and/or central heating. with correct duct placement for blowing air / return air. can make a large difference. but if the water is coming up through the concrete.... i would say more expensive route and see about getting some drainage under the floor.
if ya go with putting in drainage, you might see if you can find one of them powered 2 wheel tractors. kinda like a walk behind tiller. but you can attach different stuff to it. including a lawn cart for example. lay down some plywood on the stairs for a ramp. with some sticker traction tape. and vroom, vroom, vroom.... it would help getting the concrete out for were the drainage pipe would go. the dirt that would need to be removed, getting rock back down there, and then concrete down there to finish things out.
if you get a big enough concrete saw to cut almost all the way if not all the way through the concrete slab/floor, and make cuts, so you end up with say 12" x 6" chunks, you could use either a prybar and sledge, or an electrical jack hammer to split the chunks. were ya cut. biggest issue is getting cuts between floor you want to keep and the area were drainage would go. these cuts, ya keep jack hammer / sledge away from. so ya do not damage rest of floor. the cuts between the 12"x6" chunks, are more of a wedge/sledge hammer thing once you get things started. so you can give a slight tap "hopefully" and chunks fall out of place to be picked up, before moving on to the next chunk. ""hopefully not to much rebar to deal with""