It's still preferable to use a real DNS name and a real certificate, if you possibly can, which means that nobody has to install a CA.

There's currently no way on any modern system, as far as I'm aware, to install a CA but limit it to only a specific domain and its subdomains; any CA you install can issue a certificate for any domain. (There are theoretically some certificate extensions that allow limiting a certificate to subdomains, but as far as I can tell, implementations don't reliably support those extensions.)