The way I picture warp drive working, is that a ship carries its own inertial reference frame along with it inside the warp bubble. So if a ship at warp fired a phaser forward I would expect that the beam - while inside the "warp bubble" - would travel at C away from the ship, just like relativity predicts. But where things get confusing for me is predicting what would happen after the phaser beam passes through the warp bubble and enters a different reference frame.
, if I understand correctly you are saying that even once outside the warp field, the phaser beam will be observed to move at C away from the ship (from the ships pov).
The problem I have with that is this: Imagine that you have a radio transmitter at point A and a receiver one light-year distant at point B. Lets say that as soon as the transmitter emits a radio signal, a ship at point A goes to warp and travels toward point B. Since you say that even a ship at warp will observe light outside the warp bubble as traveling at C, this means that even if the ship is traveling, say, warp 9, the ship will not catch up with the radio signal. So how long does it then take the ship to travel from A to B?