An arcane connection to Christ

So how do player characters even notice this? If it's that hosts are being stolen... Hmm.

While I do not think the whole idea is sound I can (just) see that if hosts were stolen then the Hermetic Magi might get blamed and have to investigate to clear themselves. (Yeah, like that ever works...)

Or how about... An unexpected backlash. She tries it and pumps all her magic into affecting the Christian people of one town, the one nearest the PCs of course.

But instead of causing people to turn away from Christ she is instead pouring the magic from her regio into the innocent Christian people of the town. Who unexpectedly start to display the sort of signs you'd expect from Gifted children just coming into their power. Poltergeists, milk souring, young men growing asses ears....

The witch is caught in the focus of her blasphemous incantation, the people are panicking and the Magi are off on a quest to discover why a perfectly mundane town has suddenly as many Gifted individuals as citizens. (Apart from a couple of hidden pagans and/or Infernalsists...)

Don't think so. For crying out loud, in Ars Magica he lets infernists make Infernal Vis with it. Christ probably isn't gonna stop this either. I mean, he could, but he might also just smite the witch or negate the spells.

I for one think this could work. The Eucharist IS Christ's flesh. The Limit of the Divine protects it from Hermetic Magic. If it was merely a relic or something similar Hermetic Magic could transmute or destroy it. It might be protected by the limit of the Divine, but other magic often lacks the limits of Hermetic Magic.

I suppose this means that handing out bits of Christ to random, possibly malicious or irresponsible, peasants is a terrible idea.

Not that one. By RAW, all forms of magic face the limit of the Divine, and the limit of the Lunar Sphere. The others are negotiable, as you note.

It is. The historical Church was very careful about this: the consecrated wafer goes directly into the mouth, where saliva makes it disintegrate. It's not trivial to get hold of a consecrated host, but it is, of course, possible.