Assuming biblical accuracy ...
The answer to the secondary question is "No". He said "All Authority is given to Me." If He has all authority then this whole need to be a political activist is moot. His authority. His Kingdom.
As to your primary question ...
Take a closer look at the twelve ..
You had "Simon the Zealot" ... The Zealots were a violent Anti-Roman Jewish sect. You also had "Matthew called Levi" who was a tax collector ... a publican ... a Roman collaborator. Two completely different political extremes.
Perhaps the question should not be "Should Ministers be involved with politics"...
Observing Jesus in the Gospels, perhaps the question should be , "Does Jesus call people with strong political opinions to be ministers?"
So, the answer from scripture is "Yes".
I have found no place in scripture that said "Simon the Zealot" quit the Zealots. And I have found nowhere that said "Matthew called Levi" quit being a tax collector.
We see in the Gospels where Jesus calls people with strong political opinions to serve Him because all authority has been given to Him and He calls and chooses whomever He wishes.
It is He, not us, who determines our sufficiency as servants.