Phoenix-class Assault ship
Phoenix-class Assault carrier
to be perfectly honest, both arguments for each name have their merits, but there is a problem with "assault carrier". i just keep getting reminded of the CAS whenever someone says "assault carrier", so maybe assault ship is the way to go?
Also, in star wars, there was a ship, the Acclimator-class, that had the nickname "assault ship". It served an almost identical role as the phoenix, except it was more heavily armed comparatively