How do I launch a process with different date/time settings than the rest of the system?
No. The system doesn't bifurcate time.

You would need the app in question to handle the time shift or run independently of the system time or package the app to run in a VM or other container like docker. In some limited cases, the time zone variable might help you make a little tweak, but arbitrarily setting a totally unrelated time isn't implemented on OS X.