Your basic relationship looks sound:
Projects Contacts---<Registrations Contact ID>------Contacts
Provided that all the table occurrences down the left hand side that start with "Projects" in their names are all occurrences of the same table.
Don't quite see the need for your button here. If you base a layout on Projects Contacts, you can put a portal to Registration Contact ID on the layout and use it to assign contacts to that project by selecting contacts from the Registrations Contact ID::Contact ID field, with it formatted as a drop down list of Contact ID's and names.
Here's a demo file that illustrates this method and then also shows some more sophisticated methods fo dealing with many to many relationships: