The REST API allows you to create or modify existing sites.
https://developer.bigfix.com/rest-api/examples/create-site.html
https://developer.bigfix.com/rest-api/examples/update-site.html
The create-site documentation shows you the syntax for setting the subscription relevance for a new site. If you follow the syntax laid out in that document for the site.xml you use for your edit request, it should allow you to modify the subscription relevance for an existing one.
Disclaimer — I haven’t done this myself, but I don’t see anything on these pages that makes me think it won’t work as described.