We can do it, by using rudder to yaw into the wind direction at the start of each pass, but only holding rudder on briefly until the necessary new heading is obtained, and then promptly putting rudder back to neutral for the rest of that straight leg.
This is just a deception for the made-up rules of that particular competition. If you want to fly parallel to the runway and the wind is blowing from the right, you turn slightly right to point slightly into wind and then centre all the controls. Your comp rules won't allow you to turn properly so you have to cheat at a turn by yawing with rudder rather than banking until the same new heading is obtained as if you had done a banking turn, and then all controls to centre. No rudder, no aileron, the plane will go straight along the runway, slightly crabbing, the end effect is the same in both methods. Straight track, same heading, slightly crabbed into wind, controls at centre. The only difference is that right at the start of each pass you did a flat turn rather than a proper banked turn to set the new heading. You can not hold on the rudder, no matter how little, keep the wings level and go in a straight line, the plane will make a flat turn.