In my boat out in the Strait, I used to bonk, place in a plastic bag in the cooler (on ice), and tear/cut a gill to bleed out. The bag kept the cooler from getting too bloody. Now, I bonk and place the fish head down in a 5 gal bucket filled about 1/3 with water. After bleeding the fish goes into a bag, on ice, in the cooler. I try to clean the fish in a timely manner and leave them on ice in a new, clean plastic bag.

Not sure about a river situation, but I'd lean towards bleeding the fish in a bag in the cooler on ice.