This is a fantastic idea! The Seat of Power definitely adds more of a spin/flavor for the city's narrative than actual competition as is (and I like how a rival can essentially negate a draw's points by opposing that building's purpose too) but I love your idea of roping rivals into affecting other people's numbers! I'm planning on holding a BFB game jam for community hacks and expansions in early 2021 and would strongly encourage you to bring this idea (and more!) to the table when I do!
As promised, the #BFBJAM is now LIVE and so is an SRD for the game! If interested, I'd love to see you flesh out your Seat of Power and Rival modifications and submit them as a rules supplement! https://itch.io/jam/bfb-jam