I think the simple answer is no.
You could give managers an activity role (assign roles in the activity, not the class, but they would still need a course or category role...which complicates things a lot) of teacher or non-editing teacher or even a custom role you built which allows advanced permissions but it would not limit who they can sign up. By definition, the activity allows you to sign up/book anyone with an account (even suspended accounts), even if they are not enrolled.
There is the student signs up-manager approves it bit but I honestly don't know how it works...I've never tried it.