It looks like this thread is a little old, but in case someone else finds it useful, there are some closed-form methods for estimating N and p separately.
- Use a multivariate poisson model (Dennis et al. 2015, DOI: 10.1111/biom.12246)
- Reparameterize the likelihood of the binomial*poisson mixture following Haines 2016, DOI: 10.1111/biom.12521)
- Use distance sampling, which uses additional data to estimate p, so that you can use the count ~ poisson(mu * p) in the likelihood.