At least he did not consider an extension cord "under" the carpet. Your "bell box" link took me to a description of a "door bell box" for an anounciator type application. I would not run NM wire "exposed" under the subfloor, even if there was little possibility of anyone ever using nails or staples in the wood.
Yes, NM under the ply would be dubious. Wiremold would be pretty good stuff. If the underlayment stuff is literally 1/2" it would rule out 1/2" emt, which of course is nearly 3/4" in diameter. The NM could be run in the wiremold. There would need to be some consideration of anti-chafe at each end, but a good connector at the box would satisfy the need for a strain relief at the box. Staple it to some framing soon after it exits the wiremold and enters the wall, and that is your other strain relief.
The Bell box I referenced is one of those swell cast aluminum items one would use to support exterior lights. Specifically this thing is 4 1/4" in diameter and has threaded holes to take connectors. I know that wiremold has a coupling specifically to transition to conduit in some way or other, quite directly, but I have not worked with it.
Grainger has photos of it.
Anyway, a cast aluminum bell box screwed down to the concrete with a bronze face plate will assuredly make a solid and legitimate box to mount an outlet in. Hell, one could daisy chain and put in two, the box is plenty big enough for a line in and a line out.
The one element that may be an issue is that if the flooring under the carpet is literally 1 1/8" thick, then the box needs to rise up above that to the top edge of the carpet, or nearly. Certainly the carpet cannot be exposed to the interior of the box. And one does not want the bronze face plate to crush down the carpet noticeably.
The box could be shimmed up. Or a round face plate could close it and support the receptacle, for preference raising it about the thickness of the carpet and pad, like a mud ring. I like the mud ring idea. Get a mud ring with a suitable lift, one for a 4 11/16ths box, and bore a few holes in it such that you can screw it to the bell box. I cannot imagine why an inspector would flag it.
Or use the wire mold box with a conventional mudring of sufficient offset to lift the face of the outlet to the top surface of the carpet.
It is like an erector set: there are countless ways of getting there. And some of them are actually legitimate!