Briefly, I think you're right that there's an incompatibility between the two philosophical movements you describe. One turns on disinterested reasoning; the other denies that reasoning of that kind is possible. Be careful, however. If you mean by two different logics, two different ways of thinking through philosophical issues, then yes. I think that's right. Note, though, that many will use "logic" in the narrower sense of reasoning based on standard deductive and inductive logical principles. About it being a logical fallacy to reject an argument on the basis of the person advancing it, note that doing so is what philosophers call an "informal" fallacy. There may be nothing formally invalid or weak in the dismissive reasoning that appeals to the person, simply because there are times when the person is relevant. Sometimes genesis matters. You may have to look at the particulars to decide whether or not that's so for any given argument.
Now, the two approaches you describe can be made less incompatible by treating identitarian concerns as grounds for suspicion and heightened scrutiny but not sufficient grounds for rejecting an argument. We use genetic appeals to credentials, expertise, and track record all the time to make probabilistic judgments about claims and arguments. A well trained and experienced physician is more likely than me to be right about whether or not someone suffers from a particular illness. Albert Einstein would be more likely than me to make correct claims about relativity. But in each case, not necessarily so.
The idea that the oppressed hold a superior epistemic position to that of the oppressor is often related to Hegel's master-slave dialectic, if you want the background. But note that the principle doesn't apply to all cases. Sometimes privilege grants one a better epistemic standpoint (for example, the privilege of attending medical school), sometimes it doesn't (e.g., the blinders produced by an insular life lived only among others of the dominant class). When it comes to the question of whether identity, social position, and generally the person matters, the Devil's in the details.