Now that I think about it, my six pack cooler doesn't usually make it through the round. I use an insulated sleeve that fits in the larger pouch.
I sometimes will take a bag cooler that fits in the back of the cart basket. I throw a few waters and gatorades in it, let them check it if they want (and they usually do), then throw warm beers stuffed in my bag in the cooler after the 1st hole. It's more work, but makes doing 12+ possible.