Is there a proper way to calculate the error on the ratio of two Poisson random numbers?
As one might conclude from CDF note 6438, there is no “standard” method, even for just a single Poisson. For the ratio of Poisson’s, one can use the likelihood method to produce the uncertainty. See Joel’s calculation: it gives explicit formulae for the parabolic error and describes how to calculate asymmetric “MINOS” errors.