No they couldn’t. Fiduciary responsibility doesn’t mean you have to only do things that shareholders agree with. It means you have to act in good faith to represent their interests.
In this case Musk could easily say he had spare capacity, and if there’s crossbilling (which can happen retroactively if there was an objection) there’s really nothing to sue over. If he can get tesla engineers to try to build a cave rescue submarine to buff his public persona he can get them to do this.
As I said, I don't believe any significant TSLA holder will actually sue Musk. It's clear by now they're comfortable with Musk's view that his interests and any of his companies' interests are equivalent. But if they did, they'd have a good case unless Twitter overpaid (at which point the issue would be any Twitter shareholders).
In this case Musk could easily say he had spare capacity, and if there’s crossbilling (which can happen retroactively if there was an objection) there’s really nothing to sue over. If he can get tesla engineers to try to build a cave rescue submarine to buff his public persona he can get them to do this.