"So a Congressman who happens to be na priest has no operational role in government?"
No, I think it's not only OK, but desirable to have Priests and Ministers in government office and jobs.
I just don't believe any CHURCH should have a role or office.
A cleric with an operational role in government makes many uncomfortable. That would seem to be why the RC Church eventually directed Father Drinan to end his career in the House. I think there was only one other priest who served in the House and he was a nonvoting territorial representative for the Dakota Territory.