You're completely right. I tried, at first, to look for ways that it could be a true statement that "some areas shouldn't have consistent belief systems attached", but that made me upset or something (wtf, me?), so I abandoned that, and resolved to attack the argument, and accept it if I couldn't find a fault with it. And that's clearly bad practice for a self-proclaimed rarionalist! I'm ashamed. Well, I can sort of make the excuse of having experienced emotions, which made me forget my principles, but that's definitely not good enough.
I will be more careful next time.
EDIT: Actually, I'm not sure whether it's so cut-and-dry like that. I'll admit that I ended up rationalizing, but it's not as simple as "didn't notice confusion". I definitely did notice it. Just when I am presented with an opposing argument, what I'll do is that I'll try to figure out at what points it contradicts my own beliefs. Then I'll see whether those beliefs are well-founded. If they aren't, I'll throw them out and attempt to form new ones, adopting the foreign argument in the process. If I find that the beliefs it contradicts are well-founded, then I'll say that the argument is wrong because it contradicts these particular beliefs of mine. Then I'll go up to the other person and tell them where it contradicts my beliefs, and it will repeat until one of us can't justify our beliefs, or we find that we have contradictory basic assumptions. That is what I did here, too; I just failed to examine my beliefs closely enough, and ended up rationalizing as a result. Is this the wrong way to go about things? There's of course a lot to be said about actual beliefs about reality in terms of prior probability and such, so that can also be taken into account where it applies. But this was a mostly abstract argument, so that didn't apply, until I introduced an epistemological argument instead. But, so, is my whole process flawed? Or did I just misstep?
From your original story, it doesn't look like you have noticed that your cached belief was floating. Presumably it's a one-off event for you, and the next time you feel frustrated like that, you will know what to look for.
Now, I am not a rationalist (IANAR?), I just sort of hang out here for fun, so I am probably not the best person to ask about methodology. That said, one of the approaches I have seen here and liked is steelmanning the opposing argument to the point where you can state it better than the the person you are arguing with. Then you can exam...
Usually, I don't get offended at things that people say to me, because I can see at what points in their argument we differ, and what sort of counterargument I could make to that. I can't get mad at people for having beliefs I think are wrong, since I myself regularly have beliefs that I later realize were wrong. I can't get mad at the idea, either, since either it's a thing that's right, or wrong, and if it's wrong, I have the power to say why. And if it turns out I'm wrong, so be it, I'll adopt new, right beliefs. And so I never got offended about anything.
Until one day.
One day, I encountered a belief that should have been easy to refute. Or, rather, easy to dissect, and see whether there was anything wrong with it, and if there was, formulate a counterargument. But for seemingly no reason at all, it frustrated me to great, great, lengths. My experience was as follows:
I was asking the opinion of a socially progressive friend on what they feel are the founding axioms of social justice, because I was having trouble thinking of them on my own. (They can be derived from any set of fundamental axioms that govern morality, but I wanted something that you could specifically use to describe who is being oppressed, and why.) They seemed to be having trouble understanding what I was saying, and it was hard to get an opinion out of them. They also got angry at me for dismissing Tumblr as a legitmate source of social justice. But eventually we got to the heart of the matter, and I discovered a basic disconnecf between us: they asked, "Wait, you're seriously applying a math thing to social justice?" And I pondered that for a moment and explained that it isn't restricted to math at all, and an axiom in this context can be any belief that you use to base your beliefs on. However, then the true problem came to light (after a comparison of me to misguided 18th-century philosophes): "Sorry if it offends you, I just don't think in general that you should apply this stuff to society. Like... no."
And that did it. For the rest of the day, I wreaked physical havoc, and emotionally alienated everyone I interacted with. I even seriously contemplated suicide. I wasn't angry at my friend in particular for having said that. For the first time, I was angry at an idea: that belief systems about certain things should not be internally consistent, should not follow logical rules. It was extremely difficult to construct an argument against, because all of my arguments had logically consistent bases, and were thus invalid in its face.
I'm glad that I encountered that belief, though, like all beliefs, since I was able to solve it in the end, and make peace with it. I came to the following conclusions: