In a similar vein, I'd prefer if it were set shortly after the founding of the Federation, but in a time when things were much less tightly connected.
I'd have the Federation Starfleet essentially be a conglomeration of vessels from the fleets of member worlds (which are not yet integrated) and have it be a big deal to have exchange officers from one ship/fleet to another. The lead ship (I'd call it Phoenix
) should report to UESPA first and be sort of "on loan" to the UFP Starfleet. It makes sense after a war, with the founding of a new government, that things would be more contentious.