If they are in system they will have missions available at all stations as have a presence there, donation missions easiest way but low rep increase usually but all adds up. They might have a presence in another system as well and you can do missions there as well. Can sort BBS by Rep to see at a glance which is best for ease. If they are in control somewhere can BH at NB or Res Sites, if not KWS may add bounties for them, if they own installations may get the defend the installation missions against their enemies, if local Anarchy space will need KWS.
Yes you gain rep with side fighting for and lose rep with faction fighting against, this is normal, it will take time is all.
Megaship doesnt have missions anyway I dont think, just the station and any ground bases. BBS board on Megaship just has the factions with zero mission available last time I checked.
Ironically your best hope is if the NMLA accidentally blow up a station they are in and can do multiple passenger runs, lets hope they set the timers wrong on the next load
This wont happen so is a no go, just funny if it did. (maybe thats what happened at Lave Station)
It is a balancing act if finish the CZ, can swop every couple and it will balance out or run missions in between to keep neutral at Megaship to hand in CBs.