The positions of the splitter and boolean are asymmetric.  That doesn't imply that their moral claims on the money need be different.  It's "free money" for both, zero versus zero claim.  Zero equals zero.

On what basis do you assume that you can derive moral claims from expected outcome in an idealized, partial, and demonstrably unrealistic model of the conflict?

You lament that our wetware is obsolete and that we don't do our best to conform to an abstract, reductionist, disembodied model of rationality.  Well, we are what we are!  And the very concepts of utility and value, that are at the center of mathematical models of rationality, ultimately derive from the obsolete craves and ambitions you despise.  

What's the point of sex, 99% of the time?  What's the point of art?  Of anything, really?  You can tell yourself any story about what you want in life, you can build an intricate symbolic structure representing a rationalization about why you even bother waking up in the morning.  No matter how high and imposing the towers, that castle will ultimately rest on your mostly obsolete emotions.

Rationality helped a fair bit bring us away from the caves, but what ultimately took us from there, and what made that an improvement, were the same ultimately pointless craves and ambitions that got us in there in the first place.

So I repeat: overriding our emotions is often a requisite of rationality.  But ignoring our emotions is not automatically rational.  There must be some worthy reward, current or expected, to warrant the sacrifice.

Even if it's just "practicing restraint for when I need it."

In your case (and sorry for gratuitous speculation, but even if I'm wrong this illustrates my reasoning), accepting the $0.01 allows you to conform to a respected model of rationality and thus tell yourself, and signal to others, that you are above most of the hairless monkeys.  Bet that feels good, doesn't it?  This could be traced back to various sorts of primitive psychosocial mechanisms (self-image, self-esteem, dominance, status, ...).

In my case, $0.01 or $10 are not nearly worth the outrage of being abused.  The delta between that suffering and the satisfaction of giving greedy pig Splitter the finger is worth more than that.  Other people pay comparable amounts to get a movie or book and experience a weak version of similar emotions by proxy.  Who are you to tell me that my action is irrational?

And immoral?  My choice passes most sniff tests: Golden Rule, check.  "Think global, act local", sure.  I wouldn't mind if everyone did as me.