This was - in the issue of Simony - a central theological problem of the Catholic Church around 1050. The validity of the sacraments of simoniacal priests was rejected by Pope Leo IX in the easter Synod of 1049 - but the synod did not follow him in this, and thus it did not become canon law.
The problem got resolved by Peter Damian in the liber gratissimus: while simony is a crime and should lead to punishment of the simoniacal priest, it does not invalidate or tinge his sacraments.
By Peter Damian's argument, 12th and 13th century Catholic parishes must not fear the lack of faith of their priests. Their sacraments remain valid nevertheless - and in ArM5 their works maintain the Dominion.
Catholic priests tainting their rituals with infernal actions - like in an old Ars Magica adventure - are quite another topic in ArM5.
This best is adjudicated by the SG for each specific situation.
For sagas going several decades beyond 1220, see e. g. sub rosa #16 p.49:
After 1220 AD – the typical beginning of an Ars Magica saga – historical Europe changes rapidly: the
population increases, as do trade and literacy. People of different religions mingle regularly, haltingly transforming faith from an aspect of communities and traditions to one of individuals and convictions.