Hi, sorry for letting your question sit there for so long.
To be clear - this is the same model as discussed in Model fitting and sampling issue: Only 1 chain sampling properly - #9 by MadelineJC ? And the “one-step ahead” prediction we speak about is from this suggestion?
Anyway, this looks like quite a model, you’ve chosen a formidable foe to conquer :-) Before going deep into guesses about potential technical solutions, I think it might be useful to take a few steps back and really get to understand what each of the parameters in the ODE is doing (you might have this intuition already). The point is that maybe the problem is mathematical and it is easy to spend a lot of time fiddling around implementation only to discover that the model is not identifiable or has other issues. The choice of priors you have looks like some of the parameters might be problematic for sampling.
Could you generate a bunch of plots on how the trajectories of the ODE (ignoring the Poisson noise for now) change when you change each of the parameters? Do the parameters have biological meaning you could briefly explain? Especially, as @stijn suggested before (if I understood him right) look for degenerate cases where you get a very similar ODE solution with very different parameters (something like what I did for the sole ODE model I actually understand at ODE Model of Gene Regulation )
Best of luck with your model!