Okay so it does work and I'm an idiot. You need to allow all steam games to be installed via Steam Play in your settings and it'll let you restore them after that.
oof
Closing per the last comment.
Happened upon this thread after searching to see if this was possible; to add onto this, you seem to also need to have a version of Proton installed before you can restore a non-Proton game.
Issue transferred from https://github.com/ValveSoftware/Proton/issues/1609.
@HadetTheUndying posted on 2018-09-26T09:44:18:
Feature Request
I confirm:
contain this feature already.
Description
You can currently backup your games installed using Steam Play but cannot Restore them through the Linux Client
Justification [optional]
This is a Major Oversight and hampers low bandwidth users that take advantage of the backup system as to not go over on their datacaps. I
Risks [optional]
None, this functionality should already be present it appears to be an oversight
References [optional]
Try Backing up a Windows Only Game in the Linux Client(It will allow you.) Now try restoring it. (It will not allow you to do so.)