As soon as I enable Client Certificates for the cloud subdomain just like for other subdomains, I encounter an error:
Your web server is not yet properly set up to allow file synchronization, because the WebDAV interface seems to be broken. To allow this check to run you have to make sure that your webserver can connect to itself. Therefor it must be able to resolve and connect to at least one its trusted_domains or the overwrite.cli.url.
The setup checks run server-side. So the server will need to have a client cert issued to it I suppose and then the internal Nextcloud HTTP client will need to be configured to use it.
I assume that Nextcloud is otherwise functioning, correct? As in your clients with certs can connect to Nextcloud just fine?