That sounds like it would take more time and I do no think we have the AP to spare. We do not even live in the Empire, let the people who actually do deal with the local chaos cults.
We cannot follow every plot thread or we would not get anything done.
Given that this is the man who knows the Empress is a fraud I really, really do not think we want to do that. I would prefer both Heidi and Mandred alive over catching random chaos cultists.
Well, if you're worried about Alberich knowing too much about Heidi... there
is another issue involved here, y'know?
Look at it this way; if we capture him, then
we can interrogate him and find out
whether or not he knows the Empress is a fake Haupt-Anderssen and whether he's set up any contingencies (or just left intel caches) implicating her.
That is; the threat of Heidi's reveal might
not be ended with just Alberich's death. That's why interrogating him might be useful. Or at least -- if successful, anyway -- it might reveal to us for sure "Yeah, he
doesn't know"/"Yeah, he
didn't leave any deadrops or anything."
Choosing to kill him means deciding "Yeah there's no further threat here, if he's dead, threat over."
Because keep in mind that if we didn't stop things with
this murder, then we would have been still stuck spending AP on stopping him before the next murder anyway.
That is; some of the AP being spent is "Event AP" rather than "Main-turn AP" anyway. Interrogating Alberich might take time from "event time" rather than "turn time" so you do not necessarily have to worry about AP that much. (Though, it's also kind of uncaring to go "Meh we don't even live in the Empire most of the time anyway, doesn't matter, and I don't want to take the risk of needing to use more AP for this." Like... it
is our job and duty to fight chaos and evil for the Empire. Also, people we care about live in the Empire too.)