There aren't really too many avian specific medications generally let alone an avian specific eyedrop- which MAY be the reason why. Baytril is a common broad spectrum antibiotic used for birds with infections and there's also another one I can't quite remember now however because its specific to the eye there's probably not a specific avian eyedrop, avian medicine is unfortunately one of the least evolved areas of animal medicine purely because birds have not been domesticated as long as cats or dogs for speicific medications to be developed. Sometimes birds will have to rely on medicines for other animals so I wouldn't just stop using it- the vet may not be an avian vet but he probably knows more than anyone else you have immediate access to. Why don't you try getting an avian vet opinion via email or the internet there's a lot of avian vets willing to do this

The eyedrops will wash the infection out which is why the swelling is going down, you just need to be patient with it and see. An ointment would just sit on the eye rather than removing the infection in the same way it would protect a cut or something. With cuts, our vet uses a solution that's blue in colour and then directs owners highly diluted topical version of f10 to use to clean the wound. However this would not work on an eye either so I would stay with what the vet has given you. Tetracylcine is human grade and extremely harsh for humans!- let alone a poor bird!
Sunflower seeds with out shells will be good for now to help your bird gain weight, however this is not a good fat to have for the long term. For now this is great but I would consider making an eggfood blend too- what do you think Parajita (the eggfood would give the protein?). You can also blend safe fruits together into a puree, as well a veggies but this is more of a later stage... Eggfood is really easy to make, I do it myself because its good occassional food for birds anyway. You can get premade mix by EMP brand but I prefer to make my own

You bird still needs a little more nutrition.
With regards to not being able to leave it at the vet, well that's unfortunate. However, you will need to seek professional advice on how to feed the bird properly, the vet would have an idea about how to this even if they arent an avian vet, if not I am sure there is someone here that could give you some better pointers about weaning as you have no other choice if you can't track down the breeder.
As you said in the meantime seek out an avian vet but the vet you have now is the best hope for this bird.