A UTH won't raise the ambient temp, that is correct. I don't think I would add a screen and a lamp to the tub, as that defeats the main purpose of a tub which is its ability to hold humidity.
There are some room heaters that have a built in safety that knows if the unit is tipped over and will shut off automatically. Maybe that would be better for you?
Just curious, what are your room temps without the heater? I keep my house around 75* year round, and I have not had a problem with the lower ambient temp as long as my hot side is nice and toasty.
If your cool side is below 75, then you could use an extra heat mat or flexwatt to keep that side around 80*, it might not do much for the ambient temps, but the snake will absorb the heat from below and it will stay warm.