Could one say בימין here? And if so, is there a difference between מימין and בימין?
No, the use of בימין is mostly for political right wing (pronounced bayamin). It can also be used in construct state, like "in the right side of the house" = בימין הבית. Otherwise, the phrase בימין doesn't really exist.
This should be a reasonable alternative:
"... nimtzah betzad yammin ..."