You want to pour another slab on top of an existing slab then? And insulate between the slabs? And then incorporate radiant heat?

6" is plenty of room to work with insulation. Build up a false floor, insulate it, put in the radiant heating if you wish. Be sure to put a waterproof barrier on top of the slab, closed cell foam will be the best insulation, sub floor and flooring (or suitable base for radiant).

Not pouring anything, just building over existing slab

In case it matters, we are in the northeast so it gets cold. Our architect seemed to think he didn’t have enough space for proper installation.

Frame a 2x6 floor and insulate joist bays with fiberglass for an R19. You can foam it if you want to or some version of flash and batt (for example, foam the perimeter to seal air 0r a couple inches of spray foam and then 3.5" of fiberglass).

This ^^^ is exactly what I was going to suggest. Poly sheeting (6 mil) on the slab before the joists.

Thanks Jake. I will be sure to suggest this to our contractor (it’s Greek to me). Have you ever heard of or used Dricore?

No, I have not. It's intended for putting a flooring over a concrete floor that might get wet.

That's interesting, but if you build up the floor for insulation - and to level it - you won't be putting the flooring over concrete.

Installing a finished floor over a concrete slab can be problematic. We've done a number of screen porch to sunroom conversions that involved installing finished flooring over a concrete slab, although none were with in-floor radiant heat. What we've found to work well is to apply a waterproof coating to the surface of the concrete slab, then installing a continuous layer of rigid insulation with taped seams, followed by installation of treated wood sleepers over the rigid insulation (make sure the compressive strength of the rigid foam is compatible with the installation of sleepers) to which the subfloor is fastened. That creates an air space which can be vented to prevent moisture issues. I would avoid the use of fiberglass batt insulation due to the potential for mold/mildew growth. Make sure when fastening the subfloor that the fasteners don't penetrate the insulation or slab.