What you are trying to do is definitely unusual. The biggest disadvantage is that by introducing a discrete unobserved variable your model can no longer be sampled with NUTS alone.
Further more, conceptually, you are piling two forms of noise on top of each other (the binomial had built in uncertainty already)
I don’t understand why n would be a problem for the likelihood but not for the unobserved variable. How big of an n are we talking about anyway? The binomial should handle pretty large values.
You can also consider less flexible forms like the Poisson or more flexible ones like the negative binomial.