I, too, am a night 3 cop. I'm claiming to avoid worst-case scenarios like me checking the same person as fua, or Shirou deciding to wifom off of GL for whatever weird reason or something, mostly the former. Also to avoid the risk of my claim looking really bad if I do hit the same person as fua or if i hit the nightkill. A very important thing to note is that for scum, if a n3 cop is chaining themself to a townie, then it's just as valuable for them to kill that townie as it is for them to kill the cop; both options leave the town with one less confirmed player. And Shirou is correct that fua lock-chaining themself to Shirou is a really bad idea because it makes Shirou a really good kill for scum if he's town; it'd deny the town
With that in mind, I propose a plan. We partition the player list into 3 groups: a group that I am allowed to investigate, a group that fua is allowed to investigate, and a group that Shirou is allowed to investigate (though if he is helliishly dead-set on investigating GL then we can just split the rest of the group). The list of valid investigation targets to split between us is everyone alive except for Cephrir (because numberQ is a strictly better target) and perhaps fua (though for the sake of this process, they might as well go in one of the pools). It also might be a slightly bad idea to investigate me or Shirou simply because of the risk of us being shot and investigated, but this is a much smaller risk with us not being chained-to, so we should still be in the pools. This is basically the best of both worlds: we don't chain ourselves to specific targets so scum don't get a much bigger NK-pool, but we can guaranteedly avoid two cops accidentally investigating the same person.
In case that doesn't make sense, an example set of pools (generating these by listing players alphabetically; these are not groups that I'm proposing)
Shirou: Enchant/fua/GuiltyLion
fua: implosion/Meg/morph
implosion: numberQ/Shirou/Tejate
Each person would be obligated to investigate one of the people listed in front of them, to guarantee we don't overlap investigations.
This breakdown is also assuming we no-lim; obviously if we lim then there'd be one less person in the groups. I'm sort of indifferent between a lim and a no-lim. And if Shirou wants to chain himself to GL then his group would be just GL. I think either fua or Cephrir could propose specific groups if they want to, or we could come up with them via consensus, but I pretty strongly feel this is the mechanically best thing for us to do because it minimizes risk and guarantees that we'll get two useful cop investigations worth of info at worst (and if scum want to kill ceph, then we'd be getting *three* useful cop investigations). Unless someone has a better idea. If people feel strongly about e.g. chaining me to someone then we can do that instead.