The game runs as it would on Windows in the most recent Proton versions. Which also means that at any resolution other than 1280x720 the enemy scan text will be bugged (missing). Blame Square Enix for this, not Valve.

While you would normally use FF13Fix (https://github.com/rebtd7/FF13Fix) to solve this issue, on Linux this causes really bad stutter. However, someone posted a modified build of FF13Fix that solves the Linux stutter and makes the game playable with FF13Fix: https://github.com/rebtd7/FF13Fix/pull/102 No idea if the original maintainer will ever merge the commit but at least there is a build for now.

No other adjustments really needed. Anisotropic filtering can be enabled by using dxvk.conf - no idea where proton looks for it normally but you can force any path with DXVK_CONFIG_FILE=/path/to/dxvk.conf in the steam launch options. https://github.com/doitsujin/dxvk/blob/master/dxvk.conf

Hope this helps anyone.