I’d try to rinse the eye with saline eye wash for humans. I have small single-dose plastic vials with saline, which are easy to use for the purpose of eye rinsing, plus of course very hygienic since you only use each vial once. Wipe off the goo with clean and soft tissues/compresses, use several if you need to. Fucidine drops/ ointment is commonly used for rabbit eye infections. I have uced fucidine but never gentamicin, but gentamicin seems approved and safe for topical ophtalmic use in rabbits.
I think this is pretty much what you can do at home at this point.
An eye infection can come from an infection around a tooth root, it can be a symptom of a “rabbit cold” or it can be a local infection of the eye. If it’s about teeth or a rabbit cold, oral antibiotics should be prescribed and used in combo with eye drops. I’d also ask for an anti inflammatory painkiller to reduce pain and swelling.
Please keep us posted.