GPS receivers can do much better than 5m in terms of accuracy, even with civilian signals. But it takes time to get the intentionally-introduced jitter out. Thus, you only find them in surveying gear that has very nice antennas and very pricey boxes to do the signal processing. The military stuff is encrypted but does not feature the jitter, hence better accuracy.
I remember how during Desert Storm every sailor on the East coast rejoiced because the DoD turned off the civilian Jitter for the duration of the conflict, allowing them much better accuracy. Nothing benevolent though, just a reflection of the fact that the military was using civilian gear in Iraq because they'd run out of military GPS units and the Iraqi's didn't have any GPS gear left after day one or two of that conflict. Anyhow, sufficient time to get a very accurate GPS location is something you don't have in a rocket ascending. So with that kind of error, my suggestion is out.
Approaches with transmitters on board and processing on the ground are feasible but likely super expensive relative to the altimeter approach. So I'd try out the altimeter and see how it fares in the nose. This commercial product takes out all the fun but appears to do the job and seems to be made by this company. Nifty variations on the product, note the version that offers a 3D accelerometer. Clean USB interface with a programming application makes popping the chute at the right height very easy to program.
Such a chip by itself might be good enough for the arduino to estimate the distance traveled through multiple integration cycles. But the fact that all of their smaller gear features altimeters makes me think that a accelerometer by itself is not good enough.