try to write codes to build customized likelihood from a Normal and a Bernoulli variable by
logp_ = w1 * ln( LL(feat1) ) + w2 * ln( LL(feat2) )
It sounds like you are dealing with a mixture model, which means the data is generated from multiple distribution and mixed together.
You should instead have a look at how to model mixture model (zero inflated poisson distribution is actually one type of mixture model), you can also check related post like: