Electrical – Illuminated Mirror Wiring

It seems like the space under the cabinets is inaccessible and would be treated like the interior of a wall.

In that case, you could merely cut a hole in the wall at the back bottom of the cabinet that will hold the heater and continue your cabling into the opening that will hold the heater.

Once the cable is attached to the heater and the heater is pushed into the opening, the cable is protected.

Access to connections by removing a fixture is generally considered code compliant.

I can't quite tell from the documentation above whether the receptacle is listed to take 2 wires via clamp. When it says "either side" does it imply "both"? If not, why have 2 slots? If listed, that is a fine solution.

Pigtailing is an elegant way to do that. However a pigtail requires more box fill. Aside from having to fit as a practical matter, it counts as an additional wire for the statutory rules on box fill.

I'm a big fan of stranded wire for its flexibility (i.e. in pigtails), and I'm irked that this receptacle is listed only for solid wire. That would have to be pigtailed to solid wire if you were working THHN in conduit with stranded wire.