The shop Is correct if they are going to issue you with a Padi certification. To meet standards you would also need a Padi manual and take the Padi quizzes and final exam. This means you are repeating what you have already done with SSI.
Find another shop who will allow you to just complete your OW dives and then send you back to your SSI instructor for the issue of your OW certification.
Padi leaves it up to the individual instructor whether to participate in any cross agency referral scheme.
Personally I am both a Padi and a Naui instructor. Any referral other than Padi I wear my "Naui hat" which eliminates any issues as Naui does encourage Universal referral participation
Some of the above is not true.
From the current PADI manual, page 43:
Referrals from Other Organizations
When receiving a referred diver from another training organization for course completion, follow the Referral Procedures in the General Standards and Procedures Guide. Use the Scuba Review program and the Open Water Diver Course Final Exam to preassess the referred diver’s knowledge and skills.
From the General Standards and Procedures, pages 39-40.
When receiving a referred student diver:
1. Verify the referral documentation.
2. Before inwater activities, have referred students review, complete and sign a new:
• Release of Liability/Assumption of Risk/Nonagency
Acknowledgment Form – General Training
• PADI Standard Safe Diving Practices Statement of Understanding
• PADI Medical Statement (RSTC Medical form).
If a “yes” response in the Divers Medical Questionnaire section is different from the original form provided by the student diver, written clearance to dive from a physician is prerequisite to inwater activities.
3. In preparation for the dive and before beginning open water dive skills, assess the diver’s skills and comfort level in water and generally assess dive knowledge. If the diver exhibits lack of dive readiness, remediate before training progresses.
4. If you complete the final open water training session, ensure that all course requirements are met (including watermanship) and submit a PIC (envelope or online) or appropriate application to your PADI Office.
There is no requirement for a manual or the completion of quizzes.