Forgive my stupidty but do you mean that the system owner is irrelevant for bounty hunting?
yes. basically the system owner only defines, which ships are wanted (in its space). if you shoot only wanted ships of the controlling faction, and don't hand in bounties, the controlling faction will loose influence.
Only the issuing faction gets the influence?
correct, if you only look to the influence-gain from bounty-claims. number of transactions is the most important factor.
other factors, like ship kills apply (ship kills reduce a factions influence, no matter whether those ships are wanted or not). have in mind, that influence losses via ship kills are shared between all other factions in relation to their influence - in that way, if the controlling faction is the one with most influence, it will always get most of the influence loss of other ships by ship kills. but a single bounty transaction will weigh much more than a single ship kill.
there might be a (neglectable) influence gain via total bounty value.
I currently do BGS with the idea that only the system controlling faction bebefits from BH unless there is a station that is owned by a desirable faction.
I do BGS with a PowerPlay angle and most systems I'm flipping start out unfavourable both in terms of faction or system infrastructure. I hold my bounty bonds until such time as I get my desired party in place. maybe I don't have to?
yeah, we all did for a long time

--- it is wrong. you can cash in your bounties for the factions you want to back at which station in system you want.
i think, nobody tested bounty-transaction-numbers before, which is why we all came to the conclusion, bounty hunting helps always the controlling faction (which it did via ship kills - see above).
i took the idea totally from the 1T-trading bug plus the number of bounties claimed in the BH report.
i assume, the previous tests went like that: go bountyhunting. cash in 3 mio in bounties at this or that station. so, you will have killed around 60 ships, most of them not of the controlling faction, which will have gotten the main share of influence loss of all other factions. it will have been a single transaction cashing in those bounties. so it looked like "bounty hunting always helps the controlling faction".
i have to say, going 20 times to a RES shooting 1 or 3 small ships, and back to station to maximise bounty-hunter-transaction-number isn't what most of us understand by "bounty hunting". but it looks as if the BGS recognizes this as bounty hunting. obviously this design also favors groups (not a bad thing, i think). it also makes "system policing", e.g. interdicting wanted ships close to a station more usefull, if you don#t have a RES close by.