Hi, Kickstarter backer here, only I used my real name for my account and I'd rather not post on a forum under it.
Exactly the same thing as mentioned by the OP happened to me, and whilst I wouldn't consider myself an "RP fanatic" I have found this to be a big shock to my immersion. It might only be a little thing, but I feel like I've been railroaded and my interest in continuing to play the game has dipped. Unintended consequences are fine, but these consequences are a result of clunky game design rather than am attempt to immerse or tell a complete narrative.
I ran into this because:
The first thing which is unclear is how the Dozens are introduced to the player. "The Adventuring Hall" sounds like an utterly generic quest hub and looks like a ski chalet rather than the HQ of a religious paramilitary organisation. In the first quest you do for The Dozens you are assigned to go after an individual Knight of the Crucible by another individual for a personal reason, and in the process you expose this Knight as an imposter according to the bylaws of his organisation. This is a win-win scenario, and the main difference which emerges from this quest is that the Knights are snobs and the Dozens don't seem to do anything at all.
That this contrasts with the crazy preacher you first meet and a companion's comment that they're bigots led me to assume I hadn't worked for the Dozens but just some nice guy who was kicked out of the Knights by a fraud. I didn't actually learn anything about the Dozens, which is why I am now annoyed to learn I am locked onto that path.
The second thing which is unclear is that whilst the second questgiver makes it fairly clear that actually delivering the weapons will make the Knights angry, since you're delivering them to arm the Dozens against the Knights, nothing tells you that agreeing to look into it locks you into supporting this little theocratic militia. As others have said, this can be dealt with redesigning the quest in different ways.
But the reason why I accepted before going on to do lots of other stuff in Defiance Bay was because I was still trying to understand the politics of Defiance Bay. Whilst it's clear there's a lot of controversy over animancy, each faction's view on this controversy is absent from their first quest. Wandering around Defiance Bay solving people's quests, I very quickly understood that animancy is divisive - what I didn't understand was how this division was manifest in the politics of Defiance Bay. Even if the first quest for the Dozens had actually managed to communicate it was a quest from the Dozens, I would still have accepted the second without a warning because A) accepting quests in an RPG is like picking up keys - you always do it when you see one - and B) I didn't know enough about the Dozens themselves and their relationship to the controversey within Defiance Bay.
In short, the reason this is a problem is because this quest suffers from a combination of poor design and poor writing. This caught me by surprise as the standard for both up until this point has been excellent.
So, I'm now at a point where I am locked onto a path I really do not want to pursue because of clunky development, and reloading to a state previous would undo many, many hours of side questing. Personally, I'm going to hold off for now on either playing it or recommending it to people (which I have been doing) until we get some idea of how Obsidian are going to respond to these complaints. I'm not calling for an entire rewrite of the three factions' quest chains (although that probably would be a good thing, especially in the Dozens' case), but it seems pretty clear to me that this undesired locking is effectively a bug in that players are having poor experiences as a result of bad design. Please consider making the lock only apply after finishing the quest, not on accepting it.