Disclaimer I do not use libreelec myself (yet) but I think this may be what you're looking for: https://mullvad.net/account/wireguard-config/
From:
We provide WireGuard configuration files for any device that supports them.
So you're close, but instead of generating they private key on LibreELEC and uploading it to Mullvad, it's more like you generate it via Mullvad's web portal and download/copy it (along with a whole Wireguard config file optionally) onto your LibreELEC. Where you can then use it to configure your Wireguard.
As chewitt noted, this could be considered a security risk to let Mullvad create your private key this way, since it means their system (or at least their Javascript code running in your browser) sees/touches your private key. But all VPN providers seem to do it this way, for ease of use. Any risk is probably mitigated by the fact that presumably you trust Mullvad not to store/share your private key (just like you're already trusting them not to store/share your traffic logs or anything identifying about you), and you're not using this private key for anything else but the Mullvad VPN.
I've done it on my laptop and it worked, but I switched over to using the Mullvad app at some point (to make switching hosts and "split tunneling" (a.k.a. app exclusions) easier), so I'm not an expert at configuring/running it this way, it's been a while.
Not sure if this is what you needed, hope it helps though.