No, you sure don't want a UTH under a wood enclosure...wood insulates, & by the time you got the UTH hot enough to warm inside the cage, you'd be drying out the
wood dangerously. But you need more warmth on the floor of this enclosure, right? How about using a heat cable inside the enclosure- the kind sold to keep pipes from
freezing outside, so they're grounded...you just hook it to a thermostat, & you could probably also hide it pretty well if you want.
I've done this exact thing (years ago) to heat a tall cabinet converted for a rat snake home. I ran the cable horizontally around the back & sides, about 10" from the
actual floor. That was to leave the very bottom of the cabinet cooler, since heat rises, and that 3' cable did the job beautifully; the cabinet opened with front doors
that had glass windows. Anyway, there's many ways you can install such a cable to suit your purpose. The main thing is to have the warmth where it's needed.