Allow editing of only a single field of a node in Panels or a block
Try the Field Permissions module:


  The Field Permissions module allows site administrators to set field-level permissions to edit, view and create fields on any entity.


If you allow your users access to the group edit form, they will only see the fields that you have selected to be editable using Field Permissions.

Use that in conjunction with the form block module as you've suggested and you should get the effect that you're looking for.