Hi there!
My lop has always had a bad nose discharge but its from her horrible mouth. Have you had her nose cultured for bacteria? Has the vet done a CT to see if her teeth roots are growing into her nasal cavity? She might also have bad point on her molars that cause the sneezing and discharge.
I would just suggest a very compressive vet visit to make sure your little guy doesn’t have an underlying health issue as nasal discharge is always something not to take lightly in lops

Those are classic signs of an upper respiratory infection. Sometimes rabbits can get over them themselves, but since it has reached white-discharge phase, your rabbit needs the help of antibiotics to fight this infection.

Is there a rabbit-experienced vet you can go to to have your rabbit examined and prescribed medication? Only a rabbit-experienced vet (exotic vets, not normal ones) will know the safe medications for rabbits.

Common antibiotics for URIs include:
Oral Bactrim
Oral Baytril
Injected penicillin

… For URIs, the longer you wait, the longer treatment will take, since he bacteria are constantly multiplying. Try and set an appointment as soon as possible. Also, anything you can do to help your rabbit be less stressed is good. If he has trouble grooming off the white discharge, see if you can help with a slightly damp, soft towel. Never reuse the same towel twice, as you do not want to use a “contaminated” one. Additionally, anything to help reduce nose irritation is helpful. If his hay is dusty, try shaking it off before giving it to him.

I agree with Wick. URI would be the prime suspect. Other things like chronic irritation (e.g. tooth root problems) can cause sneezing and discharge, but I would rule out a URI first. A vet would need to diagnose the problem. If it’s a URI, they will prescribe antibiotics.

Bacterial infections come in many different forms. Your other rabbit could have it and the immune system could be better apt at keeping it down. Immune systems can be genetically stronger and weaker than others, or a period of stress can temporarily weaken the immune system. Some bacterial infections aren’t really ones to spread out to other animals until they are more severe. Lots of bacteria can cause a URI, so it’s hard to tell.

Out of curiosity, are the two rabbits both fixed and did you bond them properly? An unhealthy bond can cause more stress on one rabbit. Even unhealthy bonds can have periods of grooming and cuddling, so it can be deceptive.