Kitten has a bad URI and a swollen throat...?

I have a 4 month old kitten that has had numerous upper respiratory infections. This time is the worst so far. I've brought him to the vet 3 times in the last two weeks since the infection started. First they gave him clavamox which didn't seem to help much so I went back and they gave him more clavamox. Then I noticed his throat was swollen and his nose and mouth would bleed after he ate. When I say his throat is swollen I literally mean under his chin... not the glands. They are a little swollen too but the throat is like rock hard. Also, his eyes started producing discharge. I brought him back to the vet when I found this earlier this week and he anesthetized him to search his throat for foreign objects and swelling. He found that his throat was peeling skin which was causing the bleeding. He gave him an injection of convenia and some more clavamox. It is two days later now and the kitten's throat is still as bad as it was when I first discovered it and there has been occasional bleeding. He won't even swallow his pills anymore and as of today won't eat unless I force feed him. I think it hurts him to swallow because he won't even swallow water. I'm worried about this kitten and I feel like I have tried everything. This kitten came from a shelter and just hasn't had an easy month he has been with us. He has just been constantly sick and my heart aches for him. Does anyone have any experience with their cats getting injections of convenia? Also, I heard about people using l-lysine when other options don't work, has anyone ever tried this? Another thing, he was on steroids for a bad infection he had a few weeks ago which seemed to be the only thing that helped but the vet said that they prefer not to put kittens on steroids. Does anyone have any suggestions or insight on the matter? Any personal experience is greatly appreciated.