A special case: Dynnikov and Veselov found perfect Morse functions on $G=O(n), U(n), Sp(n)$. Since $Sp(n)$ is simply-connected, their theorem answers your question in this case. See Theorem 1.1 3) of their paper to see that all critical points of the Morse function on $Sp(n)$ have index $0$ or $>2$.
One could try to modify their technique to find perfect Morse functions on $Spin(n)$ or $SU(n)$, and thus answer your question in these cases.