I'd almost reverse the Shaw quote, because from my perspective, idealism is actually MORE conducive to working with the reasonable; if someone asks questions like "do we do X bad option or Y bad option" and forces me to make a decision, that's pragmatism, being forced to act because you just have to, can't wait for the data. I find such situations (as we're confronted with in politics every day) unreasonable, because reasonable people can easily disagree, meaning there's just no such thing as "the reasonable option."
Pure reason has no incompatibility with idealism, and in fact, this is why theoretical fields like physics sometimes work with "Idealized" conditions to shed conceptual insight on the picture in question. Things are less messy, and the math can determine everything in idealized conditions, rather than messy reality. EVENTUALLY we can subsume messy reality under purely rational frameworks, in fact the success of models in physics in totally capturing the laws of nature is testament to this...but on the other hand, when we lack the computational power to calculate every outcome, then we are forced to acknowledge uncertainty despite theoretically having control over what's going on.
In fact, sticking to working with what makes sense, what explains, rather than a "swallow the brute facts and make a decision and be tough" is precisely one variety where idealistic, impractical attitudes actually work *better* with rationality.
I think the short way to put it is there's rationality a-la pure knowledge, and then there's practical decision-making. The former can't possibly be incompatible with idealism. The latter often is.