2'6 maybe.
2'4 is a serious stretch even if you have a bat with no stickers at all,
My advice is not to concentrate too much on the dead weight of a bat, make sure it feels right. Adding grips, or lead tape around the top of the handle, or double binding the handle, or anything that adds weight in the handle area will actually improve the pickup of a bat.
Continuing on the theme of weight. I could line 10 bats that weigh 2'9 up in front of you and some will feel heavy, some will feel lighter. That's the nature of the cricket bat I am afraid. Perhaps you have a bat with a shape that doesn't aid a nice pickup, combined with a very lightweight handle. Going back to the adding weight to the handle theme, a bat with a heavier handle will pickup lighter. I am to lead to believe that handles can vary in weight by as much as 4oz.
I wouldn't say a lighter bat would assist you in playing late behind square, indeed less weight may cause you to get through the shot too early.