The Empire ships / factions aren't going to attack you simply because you are a Fed backer. I am allied with both major factions, and hold the rank of Rear Admiral in Fed and King in Empire. I travel between the two major factions with no difficulties at all (sans the typical PP interdictions). As others have said, OoM is Independent. Feel free to support the cause.
If you do BB missions for OoM, they will support our minor faction. If you participate in the war efforts in Exioce, it will support our faction.