Anecdata, for sure, but my experience working at several big companies in tech is that they won’t significantly bump your pay (and especially not your stock grant!) when they promote you. If anything, they will move you to the minimum of the salary band for the new level.
In my experience, you’re better off getting the promo and looking for the next job at your leisure. It sucks that that is what the system rewards, but I certainly don’t fault people for playing the game that is given.