What’s the size of the vectors/matrices? This doesn’t seem like an awfully large model, but the kronecker product may generate a large number of calculations if the matrices become large. Sometimes its properties can be used to avoid some of the calculations and save some time, but I’m not sure that will dramatically speed up the model.
Did you time how long a single simulation would take and/or check the message at the beginning of the MCMC run with the estimate of the time for 1000 (I think) of gradient computations to get a ballpark estimate of how long a whole chain should take?
Some models are just intense, lots of data points and/or of parameters which makes gradient calculations expensive, and sometimes the solution is just to make sure everything is working correctly and let it run for a day, a week, a month, that can happen.
You can also try a small version of the model by reducing the value of k , p, q parameters and get an idea of how the computational cost scales with size, and see if it’s worth trying to radically optimize the model, or just let it run longer.