I would actually suggest easier and safer try first a fluorescent light mounted inside the enclosure. They make about a ten degree lift to ambient air temps (surface of them run about 100ºF not a CFL just a liner tube one) So the temp is safe for momentary contact and most of the fixtures have a diffuser over the bulb dropping it down.

UTH are radiant heat sources and do not do much for air temps bulbs heat lamps and FL are convection heat so it heats air. The increase in air temps is more of what is needed rather than surface. i personally use a deep blue led over night it is a touch cooler but still in the mid 90s so it holds temps well. If you need more you can always buy a fixture for a heat lamp but typically a FL tube and fixture are cheaper than a heat lamp before you buy a bulb.