the thermostat probe needs to be OUTSIDE the enclosure sandwiched between the UTH and the OUTSIDE bottom of the enclosure. a UTH DOES NOT affect ambient temps and is not an ambient heat source; you need supplemental heating, i.e. a CHE (ceramic heat emitter) to raise ambient temperatures if you're struggling to maintain temps.
you can cover the whole thing; stagnant air won't have any affect on the snake. if you're worried about circulation, use tinfoil or saran wrap to cover the entire mesh top and cut two slits into it with a box cutter or other knife and that will allow air flow.