If the CHE is too hot for dome fixtures then I just learned something new lol. I use infrared bulbs so I'm unfamiliar with CHE but I assumed you weren't getting the full power because of the open air.
But your UTH should not get hot enough to regulate your ambient temps. It is just a source of belly heat. That's why in my BP's enclosure, the lamp and the UTH are on the same side (hot side). If you put your UTH on the hot side and your lamp on the cool side then your belly heat on the hot side is 90 and your ambient on the cool side is 90. That isn't a good gradient.