Can "not even bothering to say" be written as "nary a bother to say"?

You can say that, it sounds nice and would work in a period piece.

No. Not because the words are wrong, but the meaning is different.

nary a bother to say, means it is no trouble to say something, which is more or less the opposite of the first statement, where (regardless of how much of a bother it is) they aren't going to bother saying it.