For the sort of cooler you've got, a braid screen would work fine, although you might have clogging problems if you've got a very fine grain crush or a gummy mash. Also, a really deep mash might squash the braid closed.
John Palmer's book goes into great detail about optimum design for ice chest mash tuns. I've got a circular cooler with a "braid" (actually, several layers of stainless steel screen) wrapped around a length of slotted copper tubing which works fine.