Why not use a binomial model with a random effect over observations to account for potential overdispersion. This is conceptually very similar to the beta-binomial distribution. So if y is your response and obs is an indicator of observations (one value per observation), go for
formula = y | trials(<trial variable>) ~ <predictors> + (1|obs)
Since all the beta-binomial functions are laid out in the vignette, I have a hard time understanding you are unwilling to use it, but this is up to you of course.