We lost a lovely Amazon to a persistent infection, so I got to learn more about the subject than I would have preferred. At one time, the vet explained that his infection was located in the sinus areas near the brain, and was dissolving the bone, and that when the infection reached the brain that would be the end. And it was, the end came quickly when it came. But the vet told me he had been treating another parrot for the same infection, except that infection was localized at one of the wings, and didn't travel around. Periodic treatment of some kind kept it at bay. As I understood it, parrots have such a complex internal arrangement of hollow bones and air sacs, and fluids don't necessarily circulate through the tissues as thoroughly as in humans. So maybe that other bird's infection was located inside a wing bone? He said the antibiotics to root such a hidden infection out were riskier than managing it where it was.
One thing that did help our little guy was regular nebulizer treatments. We had a human nebulizer and a plexiglass box that fit over the perch. Set birdie in there and nebulizer some medicine, various things at various times. They breathe the medicine in deeply that way and it has a chance to circulate through some of those interior spaces, maybe. I hope your guy is on the mend and the baytril will knock the infection out for good. He sure does look at home there with that mountain of food!