Just an update here-I asked my vet about this and he said that although this would be some ingestion of the drug, it was too minimal to be a risk. That being said, my bunny has had everything but the kitchen sink thrown at him for this infection, so honestly, the risk is not necessarily outweighing the benefit of not trying. We have tried Ciprofloxacin, Gentimicin, and Neopolydex eye drops for his nose in addition to Amikacin, saline, and Gentimicin nebulization and Baytril and Orbax systemics.
A few days ago, I noticed something sticking out of his nose and pulled out a nearly 5 inch long piece of hay. Since then, his discharge has been next to none which is great, but I am not sure if this is due to the hay or Tobramycin as the hay came out about a day after I started the new drug. My main goal here is to suggest that Tobramycin could potentially be effective if anyone else has this problem. It is was they use to treat Pseudomonas in human patients, and it might be worth asking the vet about if your bunny has an infection.