Which is more important: Heeding the "demands of reason" or sticking to your values and ideals even when they demand irrational measures?
It's a false dilemma. For reason itself, that is, rationality as an existential tool, is itself a value. You have to value being rational in order to live consistently rationally. But then for what reason, aside from your basic desire for it, dictates your adherence to reason in the first place?
As Michael Heller notes in Creative Tension, for the Greeks, rationality, logic, etc., were forms of public faith, forms of public piety. Greece became what it became because it, more or less collectively, chose to put reason high in its pantheon of public well-being. The fact that not all societies did likewise, but still flourished, indicates that there is no intrinsic connection between public well-being and Greek rationality. The good man in Greek ethics should be rational. Yet, paradoxically, the good man must first value being good, which then entails being rational. Being good, however, is not a purely rational decision. It is its own category of value, and cannot be adduced from pure reason.