I think run the game under linux will not cause a ban, since Proton is shipped with Steam, and the Steam deck allows to play games that are not supported on linux, and no one was banned or whaetever. The only issue was games that required anticheat. If for some reason the anticheat doesn't run with the game you are auto-banned. But I think Valve work to fix that issue.

In the case of FFXIV is a bit different. Clean FFXIV installation should not have any issue, even Gabe Newell was playing it on steam deck. However, the official launcher doesn't work anymore on linux, so, in order to play FFXIV on linux you either need the old launcher which is no longer supported, or the unofficial FFXIVLauncher, which, well, it is a third party tool. This one is the one that can be a problem. As far I know no one was banned for use it, but I won't trust it 100% just in case.