It's up to the checker to allow something to pass or not if it's not marked with a size on it. Even if it's smaller than a marked 3 oz bottle, they may be a pain and refuse it.

I would never ask a stranger to carry stuff for me, nor would I ever do it for somebody else.

Put your carry on in one bin with a bag, and put your shoes and jacket or whatever in another bin with another bag, and the odds of them figuring they are both yours are very low.

Some quart size bags are much larger than others, if you have more than one brand/style at home check to see if any are larger.