The short version: RPing your PC is not a reason to make everyone else at your table not have fun.
I think this replaces the long winded version I have posted on the outside of my screen...
The short version: RPing your PC is not a reason to make everyone else at your table not have fun.
For those questioning this issue - this is even discussed in the Adventurers League Player's Guide on p4 and specifically on p22.
P4: No Undermining of Other Characters During Adventures.
Adventurers are brought together by common cause, and during an adventure, they’re expected to work together to overcome challenges. Though certain factions might find others distasteful, individuals will put that aside and become a team when put in dangerous situations. In short, play nice with each
other when things get deadly.
P22: How do I deal with players of evil characters, or who venerate an evil deity?
Just because a player has a character with a darker side doesn’t mean that player has a license to make the game less fun for others at the table. Players are encouraged to have their characters work together despite their differences; a little competition is fine, as long as it stays fun for everyone involved and doesn’t result in other players getting shut out of the experience. If a DM or another player feels as though a player is creating an uncomfortable situation through the excuse of “it’s what my character would do,” the DM is free to give the offending player a warning for disruptive behavior, and if it persists, ask the organizer to remove the player from the table.
tl;dr:Work together. Play nice. No infighting. "It's what my character would do" is not an excuse (regardless of alignment).
To my point above I will give an example. The 2 Zhents in our group were given a mission to poison an individual but were unable to successfully do so. Another factions mission was to talk with the individual to get some information and possible help for something else.
I generally feel CN is far more difficult than LE to deal with, but if you are having problems with the LE players, you should remind them that their factions have told them they must work with the other factions and not to cause waves or their PCs will be executed for threatening the factional alliance. The factions that allow evil PCs wield pretty big hammers when keeping their people in line. It is also important that DM's help guide/control their tables. I have seen LG paladins be just as difficult or intransigent ("Its a goblin, I put it down immediately no matter what you say because they are evil!")
As to your specific example, this appears to be a case of either a confused/unprepared DM or a DM that wanted to mix in some party conflict. The NPC the Zhents are supposed to poison (Audrec, who I note they are told about and he is generally a bad guy and is only present in the mod if there is a Zhent that needs to go after him), is NOT the same NPC the other are supposed to talk to (Bhun). If your DM set up a situation where characters were in conflict, that is unfortunate but it was nt intended.
I guess my response remains that there is no one right way to play D&D and your RPing is no excuse to make playing unfun for someone else. If people embrace those two guidelines, it will work out. There will be exceptions of course, but that's how humans are. If someone isn't fun to play with, I would encourage you to politely let them know that their actions make you uncomfortable and you are not having fun. If the problem persists and there is no remedy (they won't talk it out, the DM and organizer are no help, all the other players agree that PCs should be violent murder hobos and you suck since you want to RP, etc...), don't play with those players.
The only other option would be to remove all choice for alignment (everyone must be LG, and this is how we expect it to be played, no exceptions...). Personally (not speaking as an admin in this case), I would rather we give lots of options, expose players to diversity, and grow as a community despite the occasional bumps in the road.