Hello Hugo,
From a brief reading, it seems a multinomial, multilevel model is appropriate for your data. Discrete time points with k possible states and multiple measures within subjects. Using a multinomial likelihood, you can infer the latent subject-specific probabilities of being observed in each state, while also accounting for various fixed effects influencing transition probabilities (and potential random subject slopes as well). There is no issue with not visiting particular states, as the model will explicitly account for this without the need for zero-inflation parameters (which reflect theoretical assumptions about “false zeroes” that seem inappropriate in your case).
Here is a brief example of the syntax in brms, and here is a very thorough article explaining how to estimate and interpret a multinomial, multilevel model in the context of repeated subject measures across a fixed number of possible behavioral states.
Koster, J., & McElreath, R. (2017). Multinomial analysis of behavior: statistical methods. Behavioral Ecology and Sociobiology , 71 (9), 138.