Web Service to Update Contact Information between Two Mailing Lists

Hello,
 I’m new to using API within Qualtrics, but I’m afraid I can’t find a way to fulfill survey requirements for an upcoming project using just actions or contact triggers. 
We have two mailing lists (A and B) who take two different surveys (A and B). The contacts on these lists are paired so that Contact 1A on Mailing List A and Contact 1B on Mailing List B have the same external reference and belong to the same institution. We would like to be able to update the embedded data for Contact 1B when Contact 1A completes their respective surveys. We would use this to ensure we could customize reminders and survey text that say “[Person] at your institution has completed their survey”.
We tried using both contact and action triggers at the end of Survey A to update Mailing List B, however this resulted in creating a new contact with the same external reference and updated embedded data, rather than editing the existing contact, and required that the duplicates be merged. 
To achieve what we’re looking for, I believe we would have to include a web service block in the survey flow.
I believe I would need to use a PUT command to the URL:
https://[datacenter id].qualtrics.com/API/v3/mailinglists/[mailing list ID]/contacts/[contact id]
However, I am not sure how to find the contact id for the Mailing List B contact with the same external reference as the Survey A respondent. We have saved the recipient IDs for both lists as embedded variables such that Mailing List A includes a b_id field and Mailing List B includes a a_id field.

Example (If Person 1A has completed their survey and Person 1B has not):
Mailing List A:
Name: Person1A; Recipient ID: XXXXX1a; b_id: XXXXX1b; External Reference: 0001; b_complete: false
Mailing List B:
Name: Person1B; Recipient ID: XXXXX1b; a_id: XXXXX1a; External Reference: 0001; a_complete: true

I believe the embedded data I need to pass would look like a_complete = true, I am unsure of what query and/or body parameters I would need to include. 
Any help or advice with this would be very helpful. Thank you in advance!