Is there a non-cyclic way to handle picks?
There isn't. Even with random move order, once the game has decided priority the turn will happen in this fashion:
P1, P2, ... , Pn-1, Pn, Pn, Pn-1, ... P2, P1, P1, P2, ...
in a 1v1, it is always
P1, P2, P2, P1, P1, P2, ...