I don't use Joomla but it should be the same as Drupal / WordPress where you need to enable the permission "CiviCRM: access all custom data" for anonymous user.
Otherwise, you need to define an ACL to explicitly add the permission for everyone to the custom group containing the custom field. I feel that it's more complex and rarely needed but it ...
Just answering the question in the comments about how to export the relationship_id. One way is to run a Civi relationship report from Reports | Create Reports from Templates and include the Rel ID column:
Then from Actions select Export as CSV.
You could set up a Profile as you can specify where people go when they submit it.
therefore you should be able to set Profile A to redirect to Profile B on submission.
I expect this will work 'as is' for logged in users.
There would be a problem if you send someone to Profile A via a link with a checksum as the checksum will 'expire' before they get to ...
So I just went through and created a custom profile. I added three fields drawn from individual:
I entered a sample with a local business and the resulting contact had the appropriate relationship.
Is this what you were trying to accomplish?
AFAIK you will need to create a small extension with this functionality. You will be able to use the buildForm hook (https://docs.civicrm.org/dev/en/latest/hooks/hook_civicrm_buildForm/) to alter the list of contacts.
Have you developed for CiviCRM before? If not, I would recommend walking through the Developer Guide (https://docs.civicrm.org/dev/en/latest/)....