If batch sparging, a braid or manifold should produce the same efficiency. A manifold lends itself to fly sparging as the entire tun bottom is covered for collection. IMHO, a nice large rectangular cooler is more well suited to batching, and IME a nice sturdy braid like a water heater supply would work very well, be easy to install, and is only 10 -12 bucks. If you want something off the shelf, a bazooka screen would function just about the same, but are 2-3 times more $$.
Also, just cause you have a large cooler, don't think you need the braid to run the entire length. A short 12 - 18" sturdy braid is all you need. This thread here shows how to make a Tee. I would just run the braid straight for a rectangular cooler. I cut the braids w/ tree pruning lopping shears, and then trim w/ scissors. Also, I like to fold the far end of the braid in on itself and then crush it closed, fold in half and crush again to pinch it closed, same results, but neater w/out a frayed end.
https://www.homebrewtalk.com/f11/diy-bazooka-t-42475/
edit...if your cooler is square??? you could loop the braid w/ a tee to form a circle on the bottom. Or just run a 8 -12 inch piece of the heavy 1" braid straight out from the bulkhead.
Oh, and if you are lazy like me, but also like things to work well, here is the lazy man's ten minute mash tun conversion.
https://www.homebrewtalk.com/f11/ten-minute-cooler-mash-tun-conversion-125108/
and here is more information on installing bulkheads and braids
https://www.homebrewtalk.com/f51/cheap-easy-10-gallon-rubbermaid-mlt-conversion-23008/
All the above being said...some love copper manifolds and swear by them, and get excellent results i'm sure...just not my thing I guess.
Mike