My first thought would be a Dutch door (ie. a barn door with independent bottom and top halves), but a cat could easily scale that, and they aren't meant for screened-in situations. I've had family cats who have forced their way through locked cat-doors/flaps (screened or otherwise), so a smaller-scale locked screen situation may not work.
Do you have central AC? If so, your best solution may be to add a new vent to your reptile room. It'll be expensive, but it may be cheaper than installing a new window. I can't think of anything else beyond adding a window and a window-based AC unit.
Good luck!
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