I think you raise two separate issues : (1) the place of rationality in the acquisition of morality, let's say moral principles; and (2) the place of rationality in the application of moral principles. But first a word about rationality.
We aren't helped by the fact that 'rationality' is polysemous - many-ways ambiguous. When I think of rationality I think of having beliefs or sets of beliefs that are internally coherent and avoid self-contradiction (call this 'consistency'); of respecting inferential validity; of not asking inherently undecidable questions; of doubting and questioning and not accepting everything on authority or because it comes from a particular source; of looking for proof or evidence for what I believe or think I know; of matching the strength of a belief to the strength of the evidence; of ordering my preferences transitively (so that I do not prefer A to B, B to C, and C to A and that if I prefer A to B and B to C, I choose A over C in a situation of choice); also of adopting the means best calculated on the evidence available to me to achieve my goals or at least those goals to which I assign priority ('instrumental rationality').
▻ ACQUISITION OF MORALITY
If this is at least roughly what rationality covers and includes, I can't see that it has any direct relation to morality. How can morality be deduced or otherwise derived from it ? You would have to draw out a strand and show that, for instance, immorality involves a kind of self-contradiction. (Kant is in the wings here.) But any idea that morality is straightforwardly, without conceptual elaboration and argument, a 'consequence' of rationality doesn't seem to stand up. I do not deny that such elaboration and argument cannot be supplied - my point is that, if the idea is to be maintained, they need to be.
▻ APPLICATION OF MORAL PRINCIPLES
Now here, I think, your case is much stronger. Consider how, from the concept of justice (of treating like cases equally), we arrive at the idea that women and men should have equal pay for equal work and that women should have the same right to vote as men, that reverse discrimination might be appropriate (the favouring of groups which have been disfavoured in the past), that the demands of justice apply internationally and not only within the nation state. All these extensions of the application of justice have been driven at least in part from the rational requirement of consistency. If justice involves treating like cases equally, then we cannot consistently deny equal pay for equal work between women and men. Or to look at it another way, to deny this is to deny a valid inference from the principle of justice. This is irrational since respect for valid inference is, no less than consistency an element of rationality as set out above.