Of course it would be considered an “act of war” but as you lose on the non nuclear level in the Pacific theatre, you lose your abiity to project power there and moving up the escalation chain quickly gets you to a point where nuclear weapons may be taking out our allies or our own cities. You don’t want to get to that point.
I assume commanders in theater would have some warning that their carrier was about to be incinerated. Do they have local authority to launch tactical nukes under a use'em or lose'em rationale?