Even if we're talking about axiomatic disagreements, rational debate is still useful. Eg, we can still use rationality to help identify which axioms we're disagreeing with.

Case in point is your abortion example. I think you've messed up your lines of cause and effect there. Being anti-abortion either causes or has a common cause with believing that life begins at conception. Being pro-abortion causes or has a common cause with believing that life doesn't begin at conception.

Let me posit an axiom that causes anti-abortion. Instead of the whole 'soul' thing, lets go with "Women deserve to be punished for having sex," and that 'life-begins-at-conception' is just a rationalization. If this were true, anti-abortion should coincide with religiosity (it does) and pro-abortion should coincide with women's rights (also does). Both axioms correctly fit the existing data. How could we tell the difference... which axiom is the true axiom?

My rationalist shoes say we'd want to identify a differentiation point where these two axioms would cause different results. Have there been any occasions where "reduce number of abortions" and "punish women for having sex" come into conflict? Here's one. Turns out free access to birth control slashes the abortion rate. Less punishing women, less abortions. Cool, we've identified a point of differentiation.

Okay, so what did most of the 'pro-life' side go with? Shit, turns out they went with punishing women instead of fewer abortions and again and again and again. Well, that's not cool. For fairness' and balance's sake, I'll say that the pro-choice is probably less about integrity of body and more about wanting to fuck without consequence.

As you note, we've still got an axiomatic disagreement. In order to change the opposing side's mind we still need to shift their axiom. However, rationality has let the pro-abortion side aim their rhetorical firepower at the correct target. Instead of talking about the neural activity of fetuses, they can start making people feel more comfortable and accepting of sex. Once they're correctly targeting the true axiom, they'll have a lot more luck in shifting the opposing side's position.