You need to make the link go to your participant record page instead of the singe record page. The part that has:
Participants_Db::add_uri_conjunction($single_record_link)
Is using the $single_record_link variable which is the single record page. If you don't want to use the singe record page for the single (non-editable) shortcode, just put the [pdb_record] shortcode on that page. If you want to send them to a different page, you have to change the variable to point to that page, there are several ways to do that, but the simplest is to just put it in as a string:
Participants_Db::add_uri_conjunction('/edit-record')
Where "edit-record" is the slug of your edit record page. Change that to be whatever you use on your site.