SLC certainly could be used for this, but it wasn't clear how deep you actually need it. Over a slab, slc can go from zero to as much as many inches (depending on the type). You can typically walk on it in a few hours, but you'd want to wait at least overnight to deal with a moisture sensitive wood like in a sole plate. While the stuff has great compressive strength, I'm not sure it would hold a TapCon really well, but if you used a long enough fastener to get into the old, slab below, it shouldn't be an issue. I wouldn't use a concrete nail, though. All slc installations require the use of the brand specific primer first, though.
A much less expensive option would be to do a bonded mudbed. This is a sand/concrete mix (typically something like 3-5:1 sand:concrete) that is bonded with a slurry of cement or thinset to the existing slab, mixed to a damp consistency such that it holds together when you squeeze a clump but doesn't drip water. You'd pack it down like wet beach sand, and you can screed it off to be level. Much cheaper than slc, but takes a bit more (not much!) skill. It has good compressive strength, but not much holding power for a fastener, so the same thing would apply, use one long enough to go into the existing slab.