I've used a CHE on top of the lid, and there was very minimal heat loss. The plastic lid holds in a bunch of the heat and humidity, unlike a glass tank's entirely mesh screen. Since only a section of the plastic lid will be replaced with mesh, you're not losing as much heat as you would if you were using the tank lid.
Something about using wood near a very hot object sounds like a "no" to me. Wood is an insulator, so the "case" holding the CHE would probably get very, very hot inside if it were made of wood.
Also, if you do go with mesh on top of the lid, I'd use something harder to remove like zip-ties over glue.
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