All FW Naval aviation have procedures for engine failure the takeoff evolution. All multi engine platforms calculate the TOLD numbers for worse case engine failure (and in the case of the KC-130 two engine failure) and ability to land safely. The exceptions are the Harrier and F-35.
These procedures are practiced in the simulator at the RAG before ever getting into an actual aircraft. Engine failure during the critical phases of the takeoff were mandatory items on annual NATOPS checks up to a few years ago and I suspect are still required.
From the available information it looks eminently recoverable to the CVN, just execute the procedures as done in the simulator previously through at least one cycle.
I know lots of guys that would give themselves simulated EPs (pull power back, etc.) whenever the situation allowed; engine failure on takeoff when leading a flight until formation joined, flying simulated single engine approaches, etc.
Single engine jet? Lost engine on takeoff = lost acft.
S/F, FOG