Our house is about 22 years old. When it was first built we had occasional leaking in the basement (it's a split level so only half the "basement" is below ground level), and had an interior drain system that runs to the sump pump put in along the two exterior sides where there was a problem. This seemed to solve the water problem although the basemen has always had that musty smell. There are two bedrooms, a family room and office down there and all the spaces are carpeted. Since the floor is always extremely cold, my guess is there is no barrier and the carpet is installed directly on the concrete.

Anyways the system we had installed was suppose to be cleaned yearly (unbeknownst to us). It's been in place for 15 years now and we recently had our first issue and the carpet was soaked in several areas. The company is coming to do a clean out of the drain and I plan to toss the carpet and start over on the flooring.

And this is where I'm not sure where to proceed. For the two bedrooms, I absolutely want carpet.
Unlike most people I love the look and feel of carpeting and find tile, planks, wood, etc miserably cold and hard. The company says the carpet will have to be pulled up to clean the system. I know nothing about installing/ lifting carpeting so the idea of pulling up the carpeting and having it reinstalled yearly seems like a huge effort/expense. So we are looking at carpet tiles. Seems like a better choice than wall to wall if some get wet/damaged we can just replace/lift those. But I'm trying to figure out if I should be putting something on top of the concrete before laying down the carpet tiles. I've seen the information about the tiles with the plastic backing (such as Thermodry and other similar styles), I've seen things about the dimpled membrane, I've read about a foam barrier and then OSB on top. One consideration is there is a sofit that runs through one of the bedrooms and the current height is 6" 9" so adding much height to the floor is really going to be a problem with the height in that area (especially since one of my sons is 6'3"). I've seen about laying a plastic layer (can't remember the term now) and running it up the walls but that will interfere with access to the drain system and cleaning it.

So any suggestions, on what I can use to create a barrier between carpet tiles and concrete to give some extra warmth and prevent what I assume is the sweat buildup that creates the musty smell.