Basement – Replace portion of slab for radiant heat

Insulating out 2-3 feet won't make sense, as particularly on high desert nights far too much will be sucked right out of the extended slab to the air.

Removing the slab extension should be explored. It has no valid termite purpose, other than to provide good habitat and a staging point for termites looking to crawl up one of the inevitable cracks. French drains on the other hand help keep the soil or sandy soil dry, reducing termite habitat.

A local insulation expert will have to comment on the cost/benefit of traditional slab sidewall insulation: have them show you before & after energy statements as proof their products have a positive payback in your microclimate.

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Floor coverings are fine, but in your climate you have extreme desert temperature swings, and the thermal mass of the slab is helpful in both heating and cooling seasons.

Given what you know about how you will heat the rooms differentially, insulation is good. I'd go with mineral wool (or fiberglass) since some sound isolation is often desirable between bedroom and bathroom and foam is not great at that application.

If you will be doing the bathroom walls with a moisture barrier no additional vapor barrier on the insulation is called for.