The average influx throught the Bering Strait is around 1Sv which equates to one million m3 per second. So your hypothetical 200.000 m3/day back and forth is only about 2 m3/s!
1x2=2 2x2=4 so my hypothetical is 400,000m
3 dropping off the continental shelf every day
over and above that induced by the regular/residual current, and it's not about speed but inertia, both flows will rotate and both will have eastward inertia. If it was merely 'sloshing about' then the same water would move back and forth, it doesn't due to both it's density and inertia and this is also why there is a continuous, and I believe increasing, inflow and exchange shaping the ice in Chukchi and Beaufort.