I\'m setting up a webserver for a system that needs to be used only through HTTPS, on an internal network (no access from outside world)
You'd have to ask the typical cert people for that. For ease of use I'd get with the FQDN though, you might use a subdomain to your already registered one: https://mybox.example.com
Also you might want to look at wildcard certificates, providing a blanket cert for (e.g.) https://*.example.com/ - even usable for virtual hosting, should you need more than just this one cert.
Certifying sub- or sub-sub domains of FQDN should be standard business - maybe not for the point&click big guys that proud themselves to provide the certificates in just 2 minutes.
In short: To make the cert trusted by a workstation you'd have to either
That's all your choices. Choose your poison.