Authority to enter/edit Ward callings.

Does a Bishopric Counselor have authority in the system to enter or edit Ward callings?

If 'the system' means LCR, in the 'Other' menu near the top of the page, there is an 'Access Table' item. That will display a (large) table of functions. By selecting an entry in the 'Select a Role' selection list, you can see which functions a given calling is able to do. I see 'Manage Callings' (in the 'Membership Information' block) showing as something a ward bishopric counselor should be able to do.

If those things are not working, perhaps your calling as counselor might have been entered as a custom calling rather than a standard calling.