Bullseye can be used for "mouse fart" loads in most bottleneck rifle cases -- that is, distinctly subsonic, low-noise reloads, and like most faster burning powders is position independent -- that is, it does no harm to have a tiny charge in a big case because ignition won't be changed if the powder is all by the boolit as opposed to by the primer. Some folks prefer slightly slower powders like Unique, but there's lots of information about making very light loads with fast powders, and it's generally accepted as much safer to use a fast powder for very light loads than a slow one.
Yes, Bullseye will also work for .380 loads, or pretty much any other pistol or revolver cartridge (handgun loads are the "bread and butter" for Bullseye); you won't get maximum velocity with such a fast powder, but it'll produce loads that will cycle the action with a minimum amount of powder (i.e. lowest cost) and reduced muzzle blast compared to a hotter load with slower powder. The small amount of powder is difficult to measure accurately (0.1 grain variation is a lot in a two or three grain charge compared to a 40+ grain charge of slow powder in a rifle case), but the big issue with fast powders, in any case, is that it's possible to double charge without catching it, and a double charge is unlikely to be safe in any handgun (a double charge of a mouse fart load in a Mosin or other military type rifle might be okay, but still to be avoided if possible).