So, you blithely blew right past the bit about there being information so the PCs would know the BurgerMaster was so sensitive and likely to respond so negatively to being insulted. In that case it would be player-facing, and the players would know that.
No, I incorporated that. Unless the GM tells the players that this tidbit is fixed and immutable, it's just like saying that the knight at the gate is well known for his combat prowess, but not every doing more to quantify that. Further, there's the issues of the PCs being outsiders, not one of the Burgomaster's subjects, and that they are clearly powerful in ways the townsfolk usually are not. Saying that a person is so fixed in response that they will react in a way that may clearly cause harm to them up to and including death because they cannot abide being questioned on their ability to rule by a powerful outsider seems odd. Especially since the way Vallaki is written is almost guaranteed to drive the players into conflict with the Burgomaster.
My personal experience with this module was as a player -- a very rare treat for me. I had become very taxed at work and had little bandwidth left to GM, so one of my players offered to GM. He's not comfortable at all at off-the-cuff social back and forth in general, but can hold his own. About half-way through the first of the Baron's Sun Day festivals, as we as players began to discuss interfering with the whipping of "malcontents," he made one of the best decisions I've seen him make from behind the screen -- he dumped that storyline. He saw that it was going to go straight to hell, and he wasn't comfortable with adjudicating a highly-charged social scene, so he made the decision that it wasn't important and was better off on the cutting room floor. He asked if we'd be okay just ignoring this bit of play and letting him work on it. We were fine -- we had to go save the winery anyway. When we did return to Vallaki, the GM had altered things such that we did come into conflict with the Baron, but he wasn't at all like the write-up. Still vain, but much more amenable to compromise. Especially after his main henchman was confronted and killed in that confrontation for holding Irena captive -- something that the Baron wanted swept under the rug rather than made public.
My position is that giving the PCs information that the BurgerMaster reacts badly to being insulted, then letting the PCs succeed by insulting him (at a task other than making an enemy out of him) seems to me to violate the established fiction and to negate the consequences of the PCs' making a poor choice. Heck, I'd say the same thing if the PCs had the opportunity to learn about the BurgerMaster before negotiating with him, and didn't--that's a choice (arguably more than one choice) and there should be consequences for it. Sure, there's room for flexibility, but I think it's more warranted (in the case of insulting the BurgerMaster) if there's no way for the PCs to know before encountering him how he reacts to being insulted.
This is actually one of the biggest issues I have with how the OP was presented -- we do not know what the goal of the action to insult the Burgomaster was. I think that knowing the intent behind an action is absolutely critical to being able to properly adjudicate it. That said, the information that the Burgomaster reacts badly to being insulted would tell me that this is a challenging way to move forward, but that this information should be able to be leveraged in some way. If I try to browbeat him with insults, that's risky, but not automatic failure. But, that assumption is probably because that's how I'd do it from the GM's seat. An action to insult the Burgomaster may well even get advantage, if well framed to take advantage of the information. For example, if my intention was to enrage the Burgomaster as part of an attempt to get him off balance so I could then intimidate him, leveraging the knowledge about insults seems an excellent way to do this and thus advantage.
I also think that part of my willingness to view this the way I do is that I run social challenges under a multi-roll, multi-action framework, so a single action being hard or failing does not end the scene on it's own, unless that's the intent of the player. If you want to start a fight as a player, okay, we can do that. There are even cases where you could start a fight but not end the social engagement -- I ran that exact thing with a barfight my last session. PC was in a social standoff with some thugs, started a fight, and used a quick beatdown of the leader to move back into an intimidation to cow the remaining thugs (who would likely have overwhelmed him with numbers). Success! The barfight he started continued on without him, but the thugs that were in his way backed off to find other people to fight. In this case, I let the combat actions stand in as a success or two in the skill challenge (this was helped in that we didn't do a combat swoop into detailed combat, but instead looked at whether or not his character with the abilities he has (Gloomstalker/Assassin with Alert) would reasonably be able to defeat the thug quickly and brutally. He could, so I hinged that on an attack roll.)