The more I think about it the less I think FC mission boards are a good idea - missions all affect the BGS and if they don't it's just a pure trading tool, which we already have in place.
It would also be unfair towards those who don't own a carrier, especially if you could influence factions by remote mission access while not being in the system yourself.
Not even going to mention the programming effort required and avoiding the minefield that is potential bugs as a result of these changes.
Its already unfair though. Whether its a mission or a player flies over and makes the trade same thing (regardless of the requirements for generation).
Yeah most of them do i guess, but as long as your carrier can't be involved in powerplay it doesn't matter? If anything if you were doing powerplay you'd be selling yourself short by forfeiting the positive influence side of the reward. They could just have a null value target faction for all the missions anyway.. or just not generate trading ones.
EDIT: Before you check me, how the other missions are generated are from activity on the carrier and its worked into the upkeep.