You might want to try using a MarginalModel, which will allow you to sample models like this using NUTS. This tutorial is a nice introduction to this feature.
There also appears to be an error in your model. The breaks in the mean should be triggered by the value of t, not y.
Finally, your sampler settings are absolutely nutter butters. Asking for 7.5 million draws is insanely overkill. I would recommend not more 500-1000 draws and 6-8 chains.