This may be a bit of a detour, but if you are willing to use sampling instead of optimization (which is probably a good idea on its own if that is failing frequently), and use CmdStan instead of PyStan you will have the samples from the run(s) written to disk as they are being sampled and you will be able to access intermediate states. This would likely not be a full analysis, but neither would and intermediate optimization state, and it may serve the purpose you are seeking.
As a frequent Python user I initially resisted switching to a command line, but I realized that using PyStan is essentially creating a dictionary with data for the Stan model. With the most recent (development branch) CmdStan version you can use json files, which makes it straightforward to export Python dictionaries and pass them as arguments to the compiled Stan model (I had never used json for anything useful before). Reading the output back to Python may or may not require a couple more steps to get the same output given by PyStan, but I found it to be worth the additional control CmdStan allows.