Well, what this tells me is that your vet is only moderately experienced with rabbits, in my opinion. But what do I know. I'll just share what I've learned over the last several years, with reliable medical based references on rabbits to back it up. It's really up to you whether or not to take your vets words at face value. I've unfortunately become a bit skeptical of all supposed rabbit vets and always thoroughly question and basically interview them on their rabbit knowledge, before I decide whether or not I will trust them with my rabbits care. This is due in part to my own experience with less than satisfactory vets, and reading of others difficulties with inexperienced vets. I've gone to vets that supposedly know rabbits, only to discover that they are very much lacking. Then reading multiple accounts of people taking their rabbit to the vet and the vet not having a clue, misdiagnosing issues, and/or not prescribing the right medication for the illness.
So I'll first address ear infections. Rabbits don't always get head tilt with an inner ear infection. It can affect them in a variety of ways. Affecting their balance is one, or they might start going in circles, have facial paralysis, or just show signs of ear irritation and excessive scratching. I personally know of a rabbit that had an inner ear infection and the only signs he showed were some fur loss around the ear base from scratching, and balance issues(stumbling and falling over to the side, reluctance to hop or stay up on his hind feet). He never had any signs of head tilt. The infection won't necessarily be present in the external ear canal if it has only spread to the middle or inner ear. The infection can show up on xrays, but it also might not or could be missed by an inexperienced vet. I've known of rabbits that had to get ct-scans done to definitively determine if there was an infection present.
http://www.medirabbit.com/EN/Neurology/Otit/otitis.htm
http://wildpro.twycrosszoo.org/S/00dis/Bacterial/BacterialOtitisMediaInterna.htm
Arthritis can include the spine(spondylosis), so yes can be a cause for hind limb issues, which is what my old rabbit had. If your rabbit has actual hind limb weakness, spondylosis or other spinal degeneration is a likely cause for an older rabbit. If your rabbit can use his hind limbs but is stumbling or falling over because of balance issues, then ear infection is a good possibility because of your rabbits current eye infection issues. Though e. cuniculi, tumor, and stroke are possibilities as well for both hind limb weakness or loss of balance.
http://www.bio.miami.edu/hare/paresis.html
http://rabbit.org/hind-limb-weakness-in-the-rabbit/
http://wildpro.twycrosszoo.org/S/00dis/Miscellaneous/Osteoarthritis_rabbits.htm
http://wildpro.twycrosszoo.org/S/00dis/Miscellaneous/Spinal_abnormalities_rabbits.htm
http://www.disabledrabbits.com/paralysis.html
As for giving pen g to rabbits, yes rabbits can be sensitive to antibiotics. Some are extremely dangerous to give to rabbits orally, including penicillin, but penicillin is usually well tolerated to give by injection only as it bypasses their gut directly when given by injection, so doesn't affect it as much, usually. Giving pen g injections to rabbits is not an uncommon thing. It is commonly used to treat rabbit syphilis along with stubborn bacterial infections. There have been several RO members that have successfully treated their rabbits with pen g for various illnesses. I remember one that had a rabbit with an unknown illness, came on here and found out it was likely rabbit syphilis, talked to vet and began treatment with pen g, which cleared it up. Another rabbit had a bulging eye, due to a suspected retrobulbar abscess. They couldn't do surgery so it was suggested to try the pen g treatment(from the bicillin study). Discussed with their vet and started pen g injections, and it was successful in treating the abscess and restoring the eye to normal. There are multiple cases of rabbits being treated with pen g. Any antibiotic can be a risk to give to a rabbit. Some like the tribrissen are usually well tolerated by most rabbits, but not all. It actually ended up killing one of my rabbits due to a negative reaction to it. Pen g is usually well tolerated to be given by injection only, but there is always the chance it can affect a rabbit negatively. So not completely safe, but mostly and often used to treat stubborn head infections in rabbits.
Quote from medirabbit(
http://www.medirabbit.com/EN/Neurology/Otit/otitis.htm ):
"
[FONT="]Chloramphenicol and penicillin (bicillin) antibiotics pass the blood-brain barrier and have successfully treated middle or inner ear infection in rabbits. Trimethoprim sulfate is sometimes advised, but appears to bring poor improvement in rabbits. This could relate to the fact that half-life of this drug is about 40 min in rabbits."[/FONT]
Tribrissen is trimethoprim sulfa, so it's not always the best or most appropriate antibiotic to use. Not saying your rabbit has an ear infection or needs to be put on pen g, but just clearing up incorrect information regarding penicillin's use in rabbits. Here are some references to back this up.
http://www.rabbitwelfare.co.uk/resources/content/info-sheets/penicillin.htm
http://www.medirabbit.com/Safe_medication/Antibiotics/Safe_antibiotics.htm
http://www.bio.miami.edu/hare/jawabscess.html
http://wildpro.twycrosszoo.org/S/00Chem/ChComplex/Penicillin.htm
Don't take my word for it. Do your own research. Look into hind limb weakness, ear infections, arthritis/spondylosis, and decide for yourself. I shared the info on ear infections because of your rabbits current eye infection issues and ear infections can often be linked, but this doesn't mean your rabbit has an ear infection. It was just a possibility due to the balance issues you described. Spinal and arthritis issues are the most likely for an older rabbit. If so then you should see some improvement within a few days of starting the meloxicam/metacam, provided your vet prescribes an adequate dose. Many vets inexperienced with rabbits, do not often prescribe a high enough dose to actually help the rabbit. Rabbits metabolize meloxicam more quickly than dogs, therefore need higher and more frequent dosing. Normal dosing for rabbits is 0.3-0.6mg/kg, once daily, though split into twice daily is usually better and more effective. My rabbit was on 0.6mg/kg, split into twice daily dosing. Here are references for dosing info.
http://vgr1.com/metacam/
http://wildpro.twycrosszoo.org/S/00Chem/ChComplex/Meloxicam.htm
Good luck and I hope you are able to find the right solution for your bun, and that he will soon be feeling better.