62 gram bp can be in as small as a 6 qt, WAY smaller than your 5 gallon.
Set up is simple........hide over the UTH and a water bowl on the cool side, with paper towels as a substrate and your golden.
With the thermostat, it can't go any higher than 87*?
To raise ambient temps a simple incandescent desk lamp should work fine (40 plus watts) positioned over the cool side to middle of the tub. Raise and lower as needed to keep temps up. Try and make the lid a solid, non see through lid, because bright lights may stress out your snake.
The snake will feel warm or cool depending on what side of the cage he/she is on when thermoregulating. Seeing our body temps (surface temps) range around 80-85 (using my temp gun to measure hands and limbs) the snake will feel warm or cool to you.
Example:
Snake is on the cool side (77 degrees) and you pick it up. It will feel cool because your hand temp is warmer than the snake, being in that cool environment.