This is already a known variant (modulo the constraint on the second move not relying on the first). Best way to play it is that the white's first turn is only

This is already a known variant (modulo the constraint on the second move not relying on the first). Best way to play it is that the white's first turn is only one move, then black starts with the first double move - this is the 12* protocol (first implemented in a game called Connect6), and it practically eliminates any first move advantage