my veiled loves looking at its own reflection.
As for a factual argument, 2 sides screen and 2 sides glass would be ideal for dry and warm/cool climates as they help retain humidity.
Depending on the weather your cham will experience this could either be a burden or a blessing.
Not all chams are freaked out by their reflections too... If they do get POed, wood panels or tile or w/e they call them work great. Keeping brighter lights shining through the glass will stop a lot of fine glare, you could polorize it, paint it, or my personal favorite;
spray foam the inside of the glass then shape it it as vines and rocks and trees and paint it with pet safe paint/stains. Most people follow up by hiding the pots of real plants into the strategically thick spots, making for a very vivarium type look.
Again, if it does not cause a problem, that cage looks freaking beautiful. By all means this is a masterpiece, assuming it is as functional as it is pretty.
Spray foam is dirt cheap too compared to the cost of waterproofing such a beautiful wood cabinet.
no problems, just solutions.