I have a basin separate for washing bb clothes. To me, the bathing water is dirty becoz it's not running water. Anyway, if you feel comfortable about that, then it's not problem. Or otherwise, buy a small basin for washing bb clothes.
I will use washing machine (separate load) for bb clothes.
If there is milk on the clothes or towels (紗巾), I will ask her to hand wash with running water, or rinse the dirty clothes with running water, then put in washing machine.