User Dashboard and Edit Profile for Organization
We have used contact layout editor extension to show only necessary fields on view contact section. Using contact layout editor you can customize the view contact page. You can include profiles with limited fields and hide or remove the edit option via css or JS.
Or other option is write extension to redirect Org contact to /civicrm/contact/dashboard?cid=123&amp;reset=1
AFAIK having Relationship permission: set to view and update will only allow user to updated the related contact details but not Payments or Membership. Try giving 'Access civi membership' permission to user role