Cover the screen top with tin foil (to keep heat and humidity in) and use the CHE all the time (with a thermostat of course - either a separate one, or upgrade to one that can control multiple devices). You will need to cut out a spot for the CHE in the tin foil and will probably need to mist regularly to keep humidity up.
When you mist, always do so with hot water (tap water - 110-130F or so - never warmed up on the stove or boiling, etc). This way, when misted and the small water drops/vapor hits the air and drops temp dramatically, it will go into the cage warm, but not cold like room temp water, or worse, cold water would do. Room temp water would drop temps quickly and dramatically. Cold tap water could be catastrophic.
Remember, a UTH only creates a warm spot, it does virtually nothing for air temps. CHE's, RHP's, etc. warm up the air as well create a warm spot beneath them.
In case it's not clear, you want the following temp ranges:
Hot spot: 88-90F
Ambient temps: 80-84F
Cool side: 78-80F
Good luck and keep us in the loop.