Thank you guys for all your ideas! We were out all day doing errands and also ended up on a solution suggested by a couple people. So here's what we ended up with:

It's not exactly pretty, but darn if it doesn't work. So we have a hole cut where the lamp heats which prevents the sag and a wire lid zip tied on to help with the structural integrity so the cat walking won't effect it. Also helps because we were having trouble with the humidity being a little too much, so the hole will be great. Thanks very much for all your suggestions, I will definitely keep them in mind for future alterations!
(Ps- don't worry, all plastic edges are sanded down to a smooth edge, and there is no space for him to squeeze between the plastic and wire)