Do you have the inside dimensions of that cooler? It's important the bag touches the sides all around. A little looseness is wanted, but not too baggy, as that could make stirring more difficult, the bag's excess wanting to wrap around the paddle or long stirring spoon.
A flattened bag that's 22" wide makes her circumference (C) 44".
C = 2 * 3.14 * r
Diameter (2r) = C / 3.14 = 44" / 3.14 = 14"
250 micron = 0.25 (=1/4) mm or around 0.01 inch.
That's the diameter of the pores, yes?
I'd say pore wise that should work. But I don't know how it compares with the pores in standard BIAB bags, such as those made by
@wilserbrewer. If the pores are too tight they may plug up, not helping getting the wort out.
Maybe someone could chime in on that.
A preheated cooler should work well, yes. If outside in cold or breezy weather maybe wrap the cooler in a thick moving blanket or sleeping bag. It's quite difficult to heat up a mash that's too cold without adding direct or indirect heat.
I've done much partial mash in the beginning of my homebrew career, using a large, 2 gallon kitchen pot in a prewarmed oven. Then switched to all-grain.