It's the solid cage, I assure you. Some people can't resist putting vents in their solid cages, but that just results in an open cage. ANY cage will be leaky enough to change the air enough so there is always the proper amounts of various gasses in natural air.
"I like your light bulb idea, that will probably work great for him. I don't dare turn out his backing spot at night right now because he's weak and it gets down in the 60s at night in my house, but I'll figure something out to keep him warm enough and let him sleep well."
The basking lights MUST be dark at night. Put right next to them one or two of the dark blue bulbs, and only run them at night. This is only a temporary fix.
Hmmm...getting them to eat. As he learns o eat, he will gain weight very slowly, and igs do catch-up growth. If your ig is not sick, he just needs to be trained to eat. Put fresh food every day, and also when you go to bed, because stress might be keeping him from eating while he's visible. Stressed and newly relocated igs often eat only at night for one to three months. I assume you have access to the required varied iguana foods, which exclude spinach and reduce kale feeding to one or two servings/week. Give him banana, apple, squished berries, fragrant grapes out of a can (rinsed) along with his staple diet. Hunger will make him start to eat. He may select a favorite ood and pick it all out of his large serving. To counter that, chop his favorite very small and distribute it throughout his food.
But it sounds to me like either his food is wrong, or he hasn't learned to drink from standing water, or he is sick. SO
he also needs a vet check. His first vet check!
You MUST stop hand feeding him. He can't get enough food that way. It's more common than you might think. He needs to learn to scarf down PILES of chopped greens every day. He is naturally adapted to eat large amounts of low-nutrient leaves, and gets by very well with that. He can easily be trapped in the hand-fed scheme until he dies of something brought on by low nutrition. Just cold-turkey stop hand feeding, and make him work it out.
When you begin your non-feeding, you can use a very long skewer to spear bits of treats like banana, pizza crust, collards, 1/2 grapes, and such. Don't let him associate the served-up skewered food with the hand feeding in the near past. Put the bits of food right at the tip of his snout so he can smell them. As he learns to take food from the skewer, move the skewer closer and closer to his pile of food. He will convert to the pile on his own, driven by hunger and the new familiarity of the food you offer. \He'll eat more than you can imagine.
"As it happens, I'm a college graduate and want him to be my companion for the next twenty years" That's great, really, really, great. What a lucky iguana! He can live for more than 25 years if you mimic his natural environment closely enough and do not give him vitamin and mineral supplements. If it's a she, well, that's too much typing, and won't make a difference for the next 3 years.