I am running my plesk server with a different domain name than all other domains on the server.
How Can I create a Let's Encrypt certificate for this Domain in Plesk?
you might just create a separate subscription ( customer = none => admin subscription ), just for your Plesk-Control-Panel - domain, for example I recommend the FULL HOSTNAME of your server ( i.e.: servername.your-domain.com ) as subscription. You are then able to install a "Let's Encrypt certificate" just as within other subscriptions / (sub)domains.