Hi Matt,

I am going to give you my very honest opinion and what I feel strongly about only by the trend I have seen many times with this scenario. This is NOT a training and rehab issue, this is almost certainly MOSTLY (not entirely) a genetic issue which mostly puts it into a category of something that must be managed.

What is not a genetic issue is that, yes, most likely this behavior is being triggered by the perception that a defensive “snap” may or may not be an attack “fight”. But, the dog’s reaction to this which is to go directly into a fight and then a prey type hold is almost certainly genetic. Recognizing that is 3/4 of the solution. There is a lot of bad propaganda out there that unfairly regulates many of the bully and gladiator breeds, but at the same time many breeders and rescues do as much harm by failing to respect the breeds heritage and educate owners on the possible propensity of going into fight and prey mode more easily than the average breed when challenged.
This is not hard to breed out of a line if a breeder (back yard or not) selectively breeds against this tendency. But, unfortunately it is rare you find a breeder that understands how important it is to monitor the temperaments that are passed on. Sadly, even the dirtbag professional dog fighters understand these tendencies more so than the average unregulated breeder (as is the way in most countries). Lines of dogs are actually known right down to the genetic propensity of HOW they fight and where they aim. There are lines that are “nose dogs”, “leg dogs”, “ear dogs” etc..
Bull terriers are a breed that has only recently split from their gladiator history and in some of the eastern asian parts of the world (known as a gull terr) are still the go to dog for a strike first and strike hard fighting dog.
What is good about this dog is that he seems predictable about being unpredictable in very specific situations and as a professional that is 3/4 of the job done recognizing that and creating a plan that gives the dog a good quality life that sets him for success and keeps him and other dogs safe. Doesn’t mean necessarily that he has to wear a muzzle the rest of his life, but it will probably mean that it will be a good tool to use while finding playmates that are predictable good matches.
I agree with Teresa a command such as “leave it” (to disengage) can be useful while guiding a dog to a successful off-leash interaction (while wearing a muzzle). I have dealt with this a lot and almost always with gladiator breeds, even my own personal bulldogs and pitbulls. Our trainer Chris (who uses this forum) always has a breaking stick handy in his house just in case his two bully breeds have a conflict he can efficiently break up the fight. These are not the same as average German Shepherd or labrador conflicts.
My plan would include whoever handles the dog have a good leadership position established, formal off-leash obedience so the dog can be communicated to off the leash, and then IF the dog must have a new playmate the dog is communicated with while the dog is wearing the muzzle with the new dog. Any defensive aggression is allowed, but never anything offensive or overly assertive that ignores another dog’s attempts to defend their own body space. Dogs know the difference between goofy play bites and even defensive bites during play as opposed to non-play “buzz off!” bites that can be taken more personal.
He sounds like what we would call a “category 2” aggression case, which really is a low level issue since it is not against humans and can probably be easily managed. Sometimes pushing a dog too far causes the dog to fail and the trainer to feel like a failure. Where instead you can set this dog up for success by recognizing the genetic factor and also make it a successful placement as a trainer.

We do mainly aggression cases at our kennel and there is RARELY a dog that we “cure” of an aggression problem. It is almost always a combination of taking the dog as far as we can and then teaching the owner how to set the dog up for success through recognition of the dog’s propensities and having an easy to do management plan included.

I hope this helps, because cases like this can lead you to feel discouraged if you look at it from the wrong angle.