Hello, welcome to JustAnswer! I am Dr Ralston. Thanks for your question.
This sounds like it could be an oronasal fistula. This would be a tunnel or opening between the oral passage (inside the mouth), through the sinus, and out through the skin above it.
It would be unusual to have this happen from birth. However, it could be that a cleft palate or partial cleft palate (hole in the roof of the mouth) is present and that it has caused a communication between the outside, the sinus, and the hole in the palate.
Sometimes an oronasal fistula can form after a tooth becomes infected. The root can fill with pus and die, and the tissue around the tooth dies. The sinus then becomes abscessed and the tissue will go through the skin to get out.
All of these can be seen typically under sedation while looking at the pet. In extreme cases an x-ray might be useful or in really severe cases an MRI might be suggested to find the cause.
In all cases, infections will happen from time to time as bacteria either gets into the hole from the outside, or food and bacteria passes from the mouth into the sinus from the other side.
Surgical correction is available and often necessary to stop this chronic problem.
White pus is not usually as bad as green. It can just mean inflammation versus active infection. But the way this sounds, the smell would tell me it is infected. Antibiotics probably did work. But, it may get to a point where they won't anymore. So, surgical correction should be considered before it gets to that point.
Now, reverse sneeze sounds kind of like a person sniffing or snorting in when they have a lot of discharge in the sinus. Then, it is followed by a breathing out. So, it is sort of a see-saw noise, back and forth, like sawing a log.
It is caused by sinus or pharyngeal irritation in the back of the throat. We think that allergies are often to blame. Some dogs respond to antihistamines like benadryl. But not all.
HOWEVER, in this case it is very likely that there is a sinusitis related to what I believe could be an oronasal fistula that is causing all the signs you are seeing.
So, correct, an MRI scan would be the most informative. Lacking that, sedation and exploratory surgery might be recommended to find the hole, and close it up. That can be a complicated surgery. Some Veterinarians are comfortable with these types of surgeries. I myself have done many of these. But, others will want to consult with a Veterinary Surgeon or Dentist.
But, if present for birth, I would definitely want to get back and look into that soft palate roof of the mouth VERY far back to see if this is a cleft palate that was never diagnosed. Surgery is very likely to be successful if that is the case.
I hope that has been useful. If you have further questions, please reply and feel free to ask, and I will answer them for you.
Just Answer Expert - Dr Ralston