Ok, I'm not sure if I'm visualizing this properly, but would it be possible to cut a rectangular section out of the wood big enough to get the UTH on the bottom of the slate?
I'm not sure if I'm way off cause I'm not seeing it right in my head. But I think you could put a UTH Andre the slate. As long as you have it regulated by a thermostat it should be fine. Assuming the slate is under a half inch I think the heat would get through it just fine.
But, again I may be wrong....
I hope I'm helping