destroying ships has to be a thing that the bgs responds to or you wouldn't be able to shift it in anarchy systems the same way you shift it in other systems
Sure, but that assumes anarchy
does operate the same way as other, lawful systems.
My understanding is that you can't shift the BGS in anarchy systems the same way, for exactly these reasons; it's the commission of a crime that has the effect, within the relevant jurisdiction, not the destruction of a ship. So "murderhoboing" has no effect in anarchy jurisdictions.
If fines and bounties was the deciding factor, then doing any kind of crime (smuggling etc) and getting caught, would decrease the controlling faction influence? I'm not sure how you would confer such crime negative impacts to the other factions since what is legal or not is determined by the controlling faction. And there would be signifcantly less means of decreasing faction influence in anarchy systems.
Being caught smuggling doesn't affect because it's not a violent crime. The table FD featured in one of their BGS videos was very explicit to call out violent crime (as opposed to starport infractions, trespass or smuggling fines) only as a contributing factor for negative influence. And if it's not a crime, it's just violence, and isn't the same.
And a minor clarification... "what is legal or not is determined by the controlling faction" is a bit ambiguous. "Controlling faction" would generally refer to the
system controlling faction, but that does not determine what is legal; jurisdiction determines legality.
The system controller has jurisdiction over the whole system
except for pockets of a few Mm radius around facilities owned by other factions; in those situations the facility owner has jurisdiction. So you can definitely commit violent crime within those jurisdictional pockets and have it negatively affect that owner.
Of course, if a faction owns no assets, then they cannot be attacked this way; this is the double-edged sword of asset ownership for a faction... while you can reap all the influence gains of trade and bounties earned within your jurisdiction, you can also be attacked in ways that factions without assets cannot (and why undockable surface/space assets are nothing more than a liability, as there are no benefits to their ownership)