Or do we need a new method of rock classification for them?
The heat of atmospheric reentry and the sudden impact of, um, impact, certainly metamorphosize the minerals in a meteorite. That's a whole fascinating branch of geology. Before the metamorphosis, though, I don't think either of the other terms fit, unless you're talking about a meteor that has already been processed through a large body as lava and ejected via impact. Sedimentary might be closer, considering that asteroids do sort of aggregate together over geologic time, and there might be some volatiles involved, even some liquid H2O, if it gets big enough to heat up. Rocky, iron, and nickel meteorites might have different enough biographies that they could be different.
["The heat of atmospheric reentry and the sudden impact of, um, impact, certainly metamorphosize the minerals in a meteorite."] --
Not necessarily, only the surface would be affected (particularly on a large object), as while the meteorite becomes very hot on entry to our atmosphere, this time is relatively brief due to the high speed. In addition, the object itself is very very cold. So while the outer surface is undoubtedly "somewhat scorched, the internal structure "should" be relatively pristine. On examination of the interior, it should be possible to determine the origin (either accretion, ejection, or some other origin) for the meteorite.
Trouble is for the most part, collectors of meteorites are hesitant to allow the meteor to be cut in half or otherwise damaged. So the question continues to exist.