I wouldn't trust a cooler. I dislike the smell of coolers more than the smell of trashcans for some reason. The coolers I own, I have had for 10+ years and they still have that plastic smell to them. I think you would still have similar issues using a cooler vs a trashcan, maybe not identical issues but similar. I would just use what your have now and keep an eye out for an old acrylic tank on a local reef club or craigslist. People have been using trashcans to mix water for decades and have had good success so while we may be learning it is now not the best method to store water with it is far from the worst method. Your not going to kill your tank just because you stored water and/or mixed in a plastic trashcan.
I think it is always good to try and aerate the water as much as possible, and sealing it is the worst thing to do unless your storing it in glass containers and air tight. If you want to cover it/put a lid on it I would recommend adding an airstone/pump to it as well to keep the water aerated. Actually, I would recommend finding a way to black out the tank as well (paint, etc) if there is sunlight that gets in the garage which would lead to some algae growing in the mixing tank.