Hm. I have had skin issues (severe acne, stress/heat induces outbreaks of hives that can last a long time), and I'm a little surprised you mention tea tree oil-- it is antifungal but can make irritation (esp. dryness) worse, not better.
The concentration matters. I use a Dr. Bronner's soap with tea tree oil in my shower (adjoins the budgies' room) but it is diluted enough that I don't worry about it.
I definitely empathize with your financial situation (and with failing to get a good diagnosis, that sucks and drains your energy), but it also sounds a bit like you've decided to go ahead with this no matter what and were hoping for a second opinion that confirmed it was okay. If you truly feel like you have no other option, I would schedule the time you are around your bird so that it is when your treatment has worn off. If you put it on, do it in another room, and don't feed/interact with the bird if you can strongly smell the tea tree oil. If you put it on at night, I wouldn't sleep in the same room as the bird. If I'm applying makeup setting spray or hairspray, I do it in my car or outside our apartment on the porch (keeping it in my car actually works with my routine).
None of this is researched or professional advice! Just trying to apply common sense to your situation. Unlike certain undetectable fumes, at least the smell of essential oils is a clue to their concentration. There is a difference between me using soap with tea tree oil in the shower in a room next to my birds with the vent fan on (washes off, unlike a clinging lotion or balm), and applying the concentrated oil directly to a bird's body. Weirdly, I still see that recommended on older bird sites.