Hi and welcome to YT

. I'm so sorry it's under these circumstances - you're poor little baby

! I feel so bad for him.
So he doesn't chew/lick his paws or anything - it's just the muzzle he goes after? If so, that in itself is a little odd since almost ALL dogs w/ allergies will chew/lick their paws bc of the histamine reaction affecting their paws.
So, was a skin scraping of his muzzle done? Also, was it grown out in culture for 10 days to see if it was fungal?
Ya know, for some reason I feel like there could be something nerve related going on here bc it almost seems like EVERYthing (all of his symptoms) are focused along a certain pathway...all in the same main area (air licking, drooling, muzzle issues, nose licking etc). If a dog, for example, had some sort of tingling nerve pain, I would think the dog would try to itch at it to make it go away, ya know? So I do have to wonder about this. Ie, is something pinched from up in the neck area or something...? An xray may help.
Snoring: this could be a stand alone thing -or- could be signs of nasal structure issues, allergies, GERD (
read this), elongated soft palate, irritated nasal passages. One has to wonder if they should scope his nose to see if there is anything stuck up there too...? Something up the nose could cause nasal issues, scratching/pawing, nose licking, sneezing, drooling, stress.
Teeth/breath: the bad breath could easily mean there could be a rotting tooth issue going on. This could cause drooling, bad breath, extreme pain/stress, sneezing, swelling, pawing/scratching at muzzle. It could also lead to an oral-nasal fistula -- which would cause further symptoms and can be quite severe and painful. I would look into this as something to at least rule out as a factor. Most def put him under to get a dental done and all bad teeth removed ASAP.
Has Apoquel been tried yet for the itching? This is another miracle allergy drug, similar to Atopica. If you try Apoquel and it also doesn't work - then what that would tell me is that the itching is NOT related to the histamine/immune system reacting - and therefore, that's why neither drug could alleviate symptoms.
So, those are the things that came to the top of my mind. Don't know if any of that is helpful. I sure feel bad for both you and your precious little one!