A wise man once said, "If you find that you are digging yourself into a hole, the important thing is to stop digging."
The multiple problems you are having with the 5 gallon round igloo will likely persist with the 10 gallon round igloo regardless of the manifold you use (braid, bazooka screen, PVC or copper pipe, Phalse Bottom, etc.). Trust me, I've used the 10 gallon round cooler and have made the switch to the rectangular cooler. I've "stopped digging." I'm not going back to the Igloo cooler--it is enjoying its second life as a cooler for water and lemonade at picnics and dispensing gatorade in the dugout at baseball games. The Igloo cooler is very good for that.
The problem comes from grain bed compaction. The round igloos are only 12 inches wide max while 18-24 inches tall. There would be a lot less compaction of the grain bed if your cooler were 18-24 inches wide and the grain bed were only 8-12 inches high--like a rectangular cooler. Cut your losses--don't buy a larger round cooler just because of the false bottom you have. Get a good 82 qt (20.5 gallon) rectangular Coleman cooler (or bigger) that is "self-draining", i.e., spigot is very low in the cooler and has no hump. and buy or make a manifold.