I used medicine to treat pinkeye with Lydia a couple times, but even with the medicine, you have to apply it to the eye twice a day and keep applying for 1-2 days (I don't remember for sure) after all signs of infection disappear. So if you want to treat it with BM, I would do the same, especially keep applying once the eye clears up.

(Both times Lydia got pinkeye, I started with BM and eventually got the medicine anyway, but I don't think I was aggressive enough with the BM.)
Life with Lydia