If you unevenly apply reasoning, then you aren't being reasonable at all. At that point, you're jumping to whatever argument happens to suit your position at the time. In some cases, that may mean you jump to a position that looks reasonable. That doesn't make the actual process you used reasonable; you reached the right conclusion using the wrong method.
What's really great about reason, in the end, is that applies even in cases when you really wish it wouldn't. It forces you to agree to things when your personal gut response is "I really wish that weren't true." If people abandon reason when it's no longer convenient for them, then they subvert the chief virtue that reason offers over just believing whatever you want as it suits you.
I don't know that I agree. Positions are reasonable, not people. If a desire to be reasonable is inspired by the full moon or waking up on your left side, the resulting thought processes are still reasonable (or not I suppose if your application of reason is flawed, but not because of what made you think to be reasonable). You may not be reasonable in other cases and that decision to not apply reason equally may itself be unreasonable, but again, that's meta commentary in the context of this topic.
I would argue the basis or inspiration for holding a particular position is irrelevant to the question of validity. There's value in understanding why people think the way they do, but only for reasons other than determining the viability of any particular position.
It looks to me like you're both talking about two related but different things. Opiate is talking about how to be a reasonable person, while KHarvey16 is talking about how to address a reasonable argument. A poster's motive and balance is important to Opiate's focus, being a reasonable person. But another poster's motive and balance is completely irrelevant to the merits of their argument that you're addressing, which is what Kharvey16 is talking about. So you're both right in your respective arguments, which are not actually conflicting.
I suspect this difference of angle arose because KHarvey16 is approaching this from the perspective of a poster, who is solely dealing with others' arguments, not their character. Opiate, on the other hand, is approaching this from the perspective of a mod, who sometimes must judge the... let's call it the "aggregate quality" of a poster. Deciding whether to ban someone is generally a judgment of whether they personally have been and will be good or bad for the forum. Thus, you tend to focus not on the reasonableness of a single argument, but rather on the general reasonableness of the person making the argument.
Having said that, while I think the overall aggregate reasonableness a person's arguments is fair game for determining bans, I don't think an internal bias you infer (read: assume) based on those arguments should be grounds for moderation. In fact, reasoned and logical bias--particularly when there's an equally biased and logical opposite--is often the most effective tool for bringing to the community's attention the hard-to-miss-yet-relevant weaknesses in the opposite side's stance. In fact, our entire judicial system is based on this biased-yet-logical advocacy.
Again, I know this isn't a court room, but there are valuable principles we can derive from a system built and evolved over centuries with the sole purpose of determining truth.
We don't just have everyone (police, witnesses, etc.) hand everything over to a judge and jury, and have them parse out truth--even though a judge is extensively trained in such matters and is as close to an impartial party as we can get. Instead, to bring out all the strengths and weaknesses of both sides, we hire biased advocates trained in logic and reason to represent each side. We don't WANT lawyers to be impartial. We've found in our system that two opposing biased-yet-logical advocates is the best way to bring out truth. It's certainly not perfect, but it's the best we've found so far.
So, Opiate, I think KHarvey's argument should apply partially to what you were arguing, as well. Regardless of a person's motives and biases, if their reasoning is consistently sound in the arguments they make, they're a benefit to the community, even if the opposite side finds them annoying, and infers nasty prejudices in their character. It's always annoying to hear the weaknesses in the side you subscribe to--which is why lawyers often get on each others' nerves. But that doesn't mean hearing those weaknesses isn't good for the community--it is, as long as the weaknesses are presented with sound reasoning.