The obstacle I see as most formidable in such an undertaking is the fact that, no matter how much “rational software” our brains absorb, we cannot escape the fact that we exist within the construct of “irrational hardware”.

It's possible to design some 'rational software' that corrects for known or expected 'irrationalities' in the hardware. (A trivial example: if our hardware makes decisions by picking randomly from the set of possible decisions, software that provides a list of possible decisions ranked from best to worst will obviously fail. So the software should order all possible decisions, then make only the top of the list available to the hardware.)

As far as I can tell, being purely rational is in direct opposition to being human.

This makes no sense. It might be helpful for me to say that there's no such thing as "being purely rational": one can be a rational human, or a rational alien, or a rational computer, but one can't be a rational. It's necessarily an adjective of some condition; it's a two-place word.

The reason there is not a “School of Super Bad Ass Black Belt Rationality” could be as simple as…. It doesn't make people want to mate with you.

It probably would give you the skills you'd need to learn how to become the kind of person who people do want to mate with. If only because it would give you the skills to learn how to become the kind of person you want to be, in general.

1) Sex ... 2) Mortality ... 3) Food, water, shelter.

If these are your desires and you wish to have them, then use rationality's skills to obtain them most efficiently: polyamory, relationship skill, and birth control / protection can drastically increase the frequency and intensity of sex. Organisations like SENS are attacking the mortality problem; support them while learning about existing life extension methods like healthy eating. Determine how much food, water, and shelter you need and optimise your environment so you receive enough of each with arbitrarily high probability and arbitrarily low effort.

If being rational conflicts with my ability to provide my body with its basic needs...

Then you have named the art, and failed the 12th virtue.