I think so. The purpose of those cages are to protect your bp from the heating element. It should be constructed of material that doesn't get too hot. The clasp that holds the trap door closed should be secure. Don't get a cheap one. It's not the place to save a few bucks.
Where would you put it though? You can't put it in the middle. Your bp needs a heat gradient, a warm and a cool side. Put it in the middle and you'll heat the entire enclose. There will be no cool place for him to retreat to. Place it on the left and those ledges will make it easy for your bp to get too close to it. There is no place in the enclosure that will stop your bp from touching it with his nose. That won't hurt him, but that high ledge would allow your bp to literally wrap himself around it. That's dangerous. Install it in the right rear and you have the same problem. Just a little less likely because the ledge is smaller. If I was to put a CHE in that enclosure, I would put it in the front on the left. You still have one small branch there, but I think it's the best placement for this set up. But even if you can install it safely, that doesn't mean you should.
Have you considered using a radiant heat panel instead? CHEs can dry out an enclosure, and that enclosure has a lot of ventilation. Put a CHE in there and I'm concerned that you will have trouble maintaining humidity. I think a radiant heat panel would work better. They don't dry out the enclosure the way a CHE would. As a bonus, they don't get as hot, so they're safer.