He wouldn’t be mattir it just to absolve having to protest. As you might say, that’s a weak diyuk to make a stira between what R’ Moshe writes and what he intends.
But he does NOT mention any problem with cooking when there is a Tzorech Gadol
In 325: 10, he is only mattir carrying in a carmelis, not a reshus harabbim, although that is the classic tzorech gadol (equivilent to our case).
If you want to make the case that not having cholent or sheichar or “shaar devorim” does not meet the R’ma’s criteria for tzorech gadol to be mattir a d’oraisa, (in other words there are two levels of tzorech gadol) I could hear, but R’ Moshe doesn’t seem to hold that way.