I'm a little confused; why, if you're recirculating, is a cooler the first choice? If you wrap it with a sleeping bag or blanket, it'll hold temps within a degree or two.
I have a stainless kettle (two in fact) as my mash tun, and a RIMS controlling mash temp. I wanted to be able to dial in a specific temp, as well as do step mashes, which it allows me to do.
If you're going to go to the trouble of any sort of recirculation (and I think recirculating hot water to the inside of a mash tun is a kind of rube-goldbergish approach to this), then why not just go to a kettle with a HERMS or RIMS?
I'm not making this about HERMS or RIMS--rather, if you're going to do this, do it right, and get what you want, not the system that is cheapest and kludged-together.
My 2 cents, YMMV, void in states where it's void, and FDIC-insured.