Hi @beckyfisher,
You might want to have a look at this post and associated paper. The model was developed with survey responses in mind where participant have to indicate for instance agreement on a 0 to 100 scale. We kind of now that if the “underlying” real response is 1.3 or 2.7, people tend to just respond 0 (similar for answers close to 100). The idea of this model is that below a certain underlying threshold people answer 0 and above another threshold people answer 100.
This quote below suggest to me that this approach of a threshold might be useful to you. The explicit idea of the model is that the same underlying model is driving the 0s, 1s, and the proportions.
The paper has some comparisons with zero-one inflated beta models. I played around a bit with the model and in the thread I present what I think is a more robust parameterisation than in the original paper. I am obviously biased in my assessment!
I might have some more notes about this model and I can dig them up if they would be helpful to you.