Is this game EXPSPACE-complete?
I don't have an exact characterization but it's unlikely this problem is EXPSPACE-complete. Suppose $M^{\Sigma^*}(x)$ accepts and let $S$ be the polynomial-size set of strings queries by this machine. If I understand the game right, Alice can win by playing every string in $S$. The only way to prevent this is if $m_A$ is polynomially-bounded but that would put the game somewhere inside the exponential-time hierarchy (or lower).