Yeah, thinking on it, what even are flag bridges good for in the current formulation? They give you a (fleet) reaction bonus, but they don't really fit all too well into the rest of the system. Why not just have it so that flag bridges, rather than being "staffed" by an officer exclusively in that role, are simply manned by someone in the admin command already?
Simple idea: flag bridges, rather than being staffed by a dedicated officer, are able to be staffed by any officer in the admin command hierarchy that the ship falls under. As for the bonus... maybe something like 10% to their admin command bonus since they're now directly on-hand rather than commanding from a distance? Some sort of benefit, specific to the officer, spread across the fleet for which that flag bridge is in. Same rule about it having to be the "most senior" of all the people in flag bridges in that fleet for the bonus to apply.
Another option for the applied bonus is that an admin commander, when in range of their superior command or the headquarters itself but on a flag bridge, makes it so that the specific admin command is now centered around that ship plus 1 jump (maybe make it a tech? Or square root of naval command size?). So, let's say I have a simple navel headquarters of size 2, with a hierarchy depth of 5, and stick all those commanders onto flagships. If I fly the fleet out one system over, everything continues to work fine in the hierarchy. I can then fly out every ship except the flagship whose commander is the top of the hierarchy out one more system, and everything would continue to work fine in the hierarchy even though now none of those ships are in range of the naval headquarters. I can keep doing this and effectively chain flagships in this way, to extend out my admin command over some distance. However, the risk here is that it's rather fragile; any one of those flagships being destroyed or forced to move would destroy the entire chained hierarchy. This is more-or-less what was suggested above, except still tethered to the naval headquarters at one end.