I don't think behavioral game theory is inspired much by biology. Rather it's motivated by the discrepancies between theoretical predictions and the choices observed in lab experiments. It seeks to improve the predictive accuracy of game theoretic models by introducing behavioral assumptions that allow players to behave in ways different from the traditional Bayesian rational benchmark.
Many behavioral game theories such as level-k and cognitive hierarchy are non-equilibrium theories. This is in contrast to evolutionary game theory which studies the stability of equilibria with respect to selected evolutionary dynamics.