As trivial as it sounds, I have found that a set routine is surprisingly important. Iguanas seem to get their knickers in a knot at the slightest change.
Iguana vs cat - I'd put my money on the iguana, so please do be careful. I have two rottweilers and the iguana is the Alpha among the three of them. He doesn't care one hoot about the dogs, even when they come over to sniff and lick him, but they will get out of his way as he wanders about.
Once you enclose the rest of the cage, you may want to get a warm mist humidifier. To repeat that info here would take me another half hour, so more on the humidifier here:
Good luck with rehabilitating your boy, and I hope you come back with positive results!