The situation at worst is only a transient as the aircraft is accelerating with the "Towering" part of the take off only being enough to ensure clearing the Deck Edge and during the Towering portion there is some vision cues until passing over the edge of the deck. After that, it is Instruments only.
Just how much use is the Doppler cue for a Take Off?
Prior to the adoption of all those fancy devices, did not the RAF conduct Takeoffs without them? If the Doppler is so crucial in a Take Off however did the RAF get by in the past as they did?
We are not talking about SAR Ops and hovering about the Oggin, just departing a hell-deck from a Rig or Platform.
Your question is very appropriate since if One is obliged to control the aircraft by solely referring to Instruments exclusive of external cues then that is IMC flight even if it does not meet the definition of IMC which is based upon WEATHER criteria.
The wonderful FAA ignores that distinction as do Operators.
They will tell you to your Face the Weather is VMC/VFR because you have Ten Miles of Visibility underneath an Overcast on a Moonless Night. Granted if there is nothing to see like a ship with lights, an Oil Rig with Lights, or an Oil Platform with lights, there is nothing to be seen other than the Whites of your Eyes shining back from the Wind Screen, but to the FAA it is VFR Weather.
So I submit part of the answer lies in the definition of IMC/VMC/IFR/VFR.