Interesting question.
I could put forward some suggestions and you can reject or try them out. Firstly, regardless of any faction or commander. Listen and try to predict when artillery is going to fire. Having a rough idea of when it's coming or the spots it will aim for is going to save you a lot of units.
What I would have to say is that the Priest is a large fuel investment for the USF. If he has tanks on the field and the priest, then he's done very well and you really should be looking into aircraft to remove this. If he has gone straight into the priest with little or no tanks. You should be able to move in and take this out with little stress. AT guns and USF infantry are no match for the later game vehicles and infantry of the Axis. Running a pioneer and a tank along the edge of the map and into enemy territory can yield excellent results though you might lose your tank.
Aircraft really seems to be your best bet with this. The JU-87 from CAS or even the longer JU-87 from Lightning War, Blitzkrieg does well at taking it down or at least scattering enemy forces so that you can run a tank in and get the final kill.
With OKW I think it's a LOT more difficult to remove this unit from game play however there are some really sneaky ways you can do this using either the Luchs or JP4. Both of these units gain a stealth ability at vet2 which you can toggle. A stealth Luchs can get right into the enemy base unnoticed and deal a crushing blow to USF artillery
