As I noted in my first post, I was a little skeptical about whether a Nash Equilibrium (NE) could be evolved by taking the squared loss of each hand. My conclusions were that, given an expected value evaluation function, it was possible using a best-opponent fitness but not using a squared-loss fitness. There turns out to be a small bug in the squared loss code which caused a big change in results. Below are the results for the correct fitness function implementation.
As you can see, the fitness function does actually find the NE for the AKQ game. It stabilizes at player 1 betting 1/3 of the time with a queen and player 2 calling 1/3 of the time with a king.