The steam machine should be running pretty much the same OS as the steam deck, so it should work pretty much the same.
I got it working on my computer with Fedora linux using info from this forum's guide
(
https://www.pioneer2.net/community/threads/ephinea-pso-bb-on-steam-deck-official-thread.24850/),
then reading the comments on that guide, and then reading this guide:
https://ragol.co.uk/guides/ephinea-steam-deck-setup/
The main thing is that you'll want to install ProtonUp-Qt from Discover and then run it.
Click the add version button for either Steam and/or Lutris and select GE-Proton10-1 and then install it.
Then configure Steam (compatibility setting) or Lutris (runner setting) and choose GE-Proton10-1.
Also there's winetricks/protontricks settings to make it run in Windows 7 compatibility mode that's explained in the guides
I just noticed you commented on checking out that other guide in the comments, but this should work. Also make sure you select Direct3D 8 in the options screen when opening the game if that isn't selected.