I don't have an answer for how to update the assignments via the API, but recently created a report of which courses used the external lti Turnitin tool and how many used the Plagiarism Framework. I used CD2 to retrieve the information and then tried the API to see if it would return the information since I knew for sure which courses used either or both in assignments. The API did not return what was expected.
To retrieve courses using the Plagiarism Framework, I used a query similar to this:
SELECT c.course_id,
c.workflow_state,
c.account_id,
a.assignments_id,
a.originality_report_visibility
FROM courses c
INNER JOIN assignments a ON (c.course_id = a.context_id)
WHERE c.workflow_state = 'available' --published course
AND a.workflow_state = 'published' --published assignment
AND ((a.s_paper_check = TRUE) OR
(a.s_view_report = TRUE) OR
(a.internet_check = TRUE) OR
(a.journal_check = TRUE) OR
(a.exclude_biblio = TRUE) OR
(a.exclude_quoted = TRUE) OR
(a.submit_papers_to = TRUE))
I have a separate query that looks for the external LTI tool and joins the assignments table with the context_external_tools, and I have a regex for the URL.
While this won't help with the updating of the assignments via the API, you can pull the list of courses and assignments from CD2.