As for allowing clergy to participate in politics, this has a long history in France. There were outstanding politicians like Cardinal Richelieu in the past, and now there are active ministers like Brian and Talleyrand.
So, assigning some additional official positions to the Church would not have any major impact on France's political scene.
In fact, historically, clerics who entered politics almost always became pure bureaucrats and gradually distanced themselves from the Church—after all, "where one sits determines one's standpoint."
Moreover, this batch of clerics entering politics would inevitably lead the way in furthering their education at the Administration College—their own Church-established college; how could they not support their own institution?
Archbishop Beaumont, hearing that official positions could be "allocated" to the Church, immediately became full of smiles again, "Helping His Majesty the King govern his subjects is also a duty of the Church."