I have occasionally gotten into arguments about what artistic sensibilities should be encouraged, but those are generally moral arguments that happen to deal with artwork (and in this case I do worry about it as much as I worry about morality at all).

Yes, I have had in mind such arguments, and even more simply arguments over what's beautiful. I wouldn't classify them under morality label, but if you do, and are consistently worried about them in the same way as about other moral arguments, fine with me. I appologise for expecting that you are less consistent than you really are.

I would like to be able to argue with people about what the correct decision is, even and especially on hard problems which aren't instantly solved by my intuitions.

The problem I have with this is: why should I care about situations where my moral intuition doesn't give anwers? For me, morality is sort of defined by strong feelings associated with certain behaviors, and desires to reward or punish the actors. In absence of these feelings I simply consider the question morally neutral. That's why I have mentioned the aesthetics. I have no consistent idea of what is beautiful. Although it is important for me to live in a beautiful environment, clever arguers can probably change my perception of beauty a bit. Now, if I had a logical system built partially on my aesthetic intuition, but consistent and stable, I could profit as much as you in case of morality. I could algorithmically decide whether something is beautiful or not. But if I imagine encountering something new, unexpected and ugly, and in the same time calling it beautiful because that was the output of my algorithm - well, that seems absurd. (I like to think this was what went wrong with many of the modern artistic styles, like brutalism. They created a norm and later decided what is beautiful using the norm.)