Drive is undefined, so I choose to drive it with a constant voltage; setting the voltage to zero to turn off the relay results in exactly zero flyback in the circuit.

What is a "fixed core" relay, anyway? All the relays I can think of are reluctance based... Magnetized latching relays, perhaps?
In any case, the mechanics almost always move slower than the electrical circuit -- and if you're concerned about flyback, that implies an unclamped switch which will discharge quite quickly -- so it makes little difference what the core is and how much it's moved, because it's moved hardly at all in that time.
The most common case where it does matter, is the worst case, when using a diode clamp (or as above, with a clamped zero coil voltage). Here the turn-off is limited by the L/R time constant of the coil itself, and contact opening can be several to tens of milliseconds later than it would be otherwise. That is, the mechanics are actually moving while the coil is still discharging. But, this only happens because the flyback is (aggressively) clamped in the first place...
In time-critical applications, a modestly clamped flyback voltage is a good idea. Practical example: automotive injectors. They're also lower mass, so the electrical and mechanical time constants can overlap.
So, to summarize --
1. Driver isn't defined, so we have no idea what the flyback might be, if any at all.
2. The armature probably isn't moving much during discharge anyway, so there isn't likely to be a difference due to core, if the electrical parameters are the same.
Tim