Oh, I didn’t look careful at the example data. You know the number of males and female voter, but not how many each candidate got from each…
Yeah, it sounds like you would like to have is a convolution of Binomials, or a Poisson Binomial, which is not really tractable for your large N.
Since you have a large N, and possibly not extreme p, you could use the Normal approximation to the Binomial, and then the convolution of the two normals has a closed form solution.
This is probably close in spirit to your idea of modelling p as the gender ratio (I don’t know exactly what you had in mind there though).