It appears that a custom field with a permission that sets the role to "run" for reporting disables viewing that field in system notes for the role that only has "run" access. Does anyone know of a work around for this?

Why not give that role edit access to the field via search? it doesn't mean they can edit the field - just that they can use it a search parameter.

It actually has edit access through the UI, but users should only be able to see data tied to themselves.