"IMO the gaps between the rationals are small enough that it doesn't matter if you can prove this for irrationals"
Excuse me, but your opinion is wrong. Rational numbers are said to be sparse in the real number space. For the argument see "Lebesgue Measure." As for why there are more irrational numbers than rational numbers see "Cantor's diagonal argument".
Your reasoning is however correct. If P(HEADS) = p, P(TAILS) = (1-p). The probability for coin tosses are:
HH = p*p
HT = p(1-p)
TH = (1-p)p
TT = (1-p)(1-p)
Eliminating HH and TT leaves HT and TH at p(1-p) probability. There's no assumption on p being rational or not. However the further you are from p=0.5, the longer it takes to get a "valid" flip.