Hello, my name isXXXXX and I have over 20 years of experience as a veterinarian.
I am sorry to hear that your fellow struggles to breathe and salivates with exercise so much so that he gags and vomits.
Dog allergies tend to cause itchy skin rather then nasal congestion, so I would be less inclined to believe that this is allergy related.
While metabolic organ disease (kidney or liver disease or pancreatitis) are possible causes of vomiting he is fairly young and he only vomits with exercise so they seem less likely.
Given his young age I would be more concerned that there is some sort of obstruction in his nasal passages that makes breathing difficult for him or an overlong soft palate that interferes with airway exchange. Dogs that are anxious drool more and increased amounts of saliva will only make breathing more difficult, especially if he is trying to mouth breathe. When he is gagging and struggling to breathe that stimulates him to vomit.
The other possibility is that he has a weak gastroesophageal sphincter and is experiencing reflux such that when he exercises painful stomach acid is splashing up into his esophagus, pharynx and the back of his nasal passages causing increased drooling and nasal mucous production. These pups also vomit easily because of the poor function of their sphincter, which further irritates the pharynx and esophagus. Bulldogs are prone to a herniation of the stomach into the esophagus called a hiatal hernia, and that too will lead to reflux.
Ideally your fellow wound be anesthetized and have his nasal passages scoped, his soft palate examined, and his esophagus scoped to look for signs of acid burns in his esophagus as well as to look at the functioning of his gastroesophageal sphincter. Of course before anesthetizing him they will want to check some blood tests to make sure his internal organ function is normal. They may also want to take some radiographs to look for an abnormal esophagus.
In the meantime at home in case this is related to poor gastroesophageal sphincter function you can try either:
1) Pepcid-ac (famotidine) at one 10mg tablet per 9 to 18 kilograms of body weight every 12 hours.
2) Prilosec (omeprazole) at one 10mg tablet per 9 to 18 kilograms of body weight every 24 hours.
In his case I would choose Prilosec to try first because it has weak pro-motility effects as well as reducing acid production so it may be more helpful for him. These will reduce stomach acid and should help if this is related to reflux. These medications are quite safe and can be used long term if necessary.
I would also start feeding a temporary bland diet mix of 1/3 boiled, lean hamburger (or boiled, white, skinless chicken), all fats and juices drained off the meat, mixed with 2/3 boiled, plain white rice. Feed several small meals a day. If the diet seems to help you can feed a prescription low irritant balanced diet called Hills I/d long term.
These are temporary measures to at least help him feel more comfortable until we can look further into the cause of his symptoms.
Please let me know if you have any further questions.