Been a while since I watched FC. Do they explicitly say that the Phoenix used a M/AMR? If not, I don't see a need for one of those either. A short flash of warp jump might just need a big enough battery. Remember, the in-aptly named "warp core" is really just the power supply to the warp engines. The closest to official explanation in the TNG:TM would have us believe that the M/AM reactor simply generates the wattage to power these huge warp engines. The engines are the critical part to warp speed, not where the power comes from. There's nothing magical about the energy so generated, it's just happens to be the most efficient way to make that much sustained power in a ship-portable system. Enough D-cells would do the job too, if only for a short time.
Also, let's not compare these engines to a runabout's. A runabout is capable of high warp speeds sustained for weeks at a time, where the Phoenix broke light-speed for a few seconds. Hardly comparable.
--Alex