Regarding the vet:
A dragon can care for itself if being healthy and provided with anything they need. If they're seriously ill, they will not heal by some "self-healing superpowers" but will get more ill and die like any animal. (In the wild, only approx. one out of 100 baby bearded dragons reaches adulthood - the other ones are either eaten by other animals, or die because of illnesses from such as parasites, and injuries from such as accidents or when barely escaping a predator and being hurt.)
My dragon has never been to the vet as luckily he's healthy so far (behaving and looking normally). If anything is off, and nothing that can be done at home is a good solution, I'll go to the vet with him.
Which kind of lamps are you using? (Is this a lamp like a coil? Do you have one that is long like a tube?) Is there any of the mesh fabric under the lamp(s)?
The mesh on the sides: If that's fabric and not metal, it's really not good as your dragon's nails and fingers get stuck in that and he hurts himself trying to free his nails and fingers. A metal mesh like for a bird cage could work, but glass on the front, and on the sides either glass or MDF is better as with that you can keep the temperature high inside the enclosure. Do you measure the temperature inside the enclosure, could you let us know the temperature values?
In total, looking at the cage, I would guess you can improve it without the need of getting a completely new one. Likely one of the lamps needs to replaced by a better one. (With the wrong type of lamp, the spectrum will be very different from what natural sunlight is. This means that then his body cannot metabolize his food and the calcium supplement he needs correctly, leading to huge problems with bones, nervous system, skin, organs and even eyes. A dragon will get seriously disabled and die from that over time. With the right type of lamp however, this will work and no such health issues occur.)
Could you maybe take a photo from a bit away, maybe some 2 or more meters away?