You can do all of this without any problems. It will work better than you expect. Instead of enabling remote on the new server, just enable Tailscale on it. Set it up like a normal dvr with the hdhr. Now you can use custom channel source in your home server and import the m3u playlist from the remote server via Tailscale ip. This will allow you to watch, record, and integrate the remote hdhr channels right in your guide. No need to fiddle with changing servers to watch/record remote hdhr. The remote server relatives can record and they can only see their recordings and you will only see your recordings from your server. Basically two stand alone dvrs that share a single hdhr source.
When setting up I would also add Tailscale to the machine for remote access for administrative and maintenance, in addition to enabling Tailscale within the channel’s settings. Basically you will have two Tailscale ip’s. One for channels and the other for ssh like Termius app and VNC viewer app for remote screen viewing if you go with a desktop Ubuntu version.