Connecting FC UC to the BGS in any way would be a terrible idea. It would result in withdrawing more strategic elements from BGS. Stations are strategic assets in BGS for a reason. Owning / acquiring a station in a system or distributing them among factions is a strategic advantage (or, to a lesser extent liability). FCs providing certain services and large landing pads subtracts from the strategic value of stations, but only a reasonable amount (that might be still decisive when two factions compete). UC and trade for BGS would however nullify the strategic value of stations, again subtract from the strategic element of BGS and would annul years of efforts of factions building defense or ensuring certain resilience in/of their systems.
I think the short and curlies of it is, FC's are
commander assets and, in part, Squadron assets. Not faction assets. Therefore, interactions with FCs shouldn't affect Factions.
Bounties and Bonds are the only time I'll waver on that (and I believe, they do count) because they're targeted to the faction that issues them, not by the player, and frankly, BGS influence should count kills, not bounty/bond hand-ins.