Thanks so much for intending to send this as a PR!
I don’t think that your distribution’s parametrization is wrong because it has discrete random variables, that’s just the way it is. Sampling discrete RVs has the difficulty of not being tractable with HMC because there are no gradients, so all discrete distributions suffer from the downsides of the Metropolis step method. With luck, in near future a novel sampler that mixes well discrete RVs will eventually come out, and will make everything better. There are some papers on the topic.