One tough uphill battle currently facing Steam Play Proton and the Steam Deck is still anti-cheat. While there has been improvements, it's still not an ideal situation but at least one more developer is looking into it with Bohemia Interactive for DayZ.
It's a little different to support native Linux builds of games, as developers would use the native Linux versions of anti-cheat but for games that don't support Linux like that, they will rely on the Steam Play Proton compatibility layer to run the Windows version. The problem there remains that with Easy Anti-Cheat developers need to upgrade their SDK, then of course it all needs ensuring it works correctly. With BattleEye previously mentioning what sounded like the same.
So recently a user asked Bohemia Interactive about DayZ (Reddit), and the reply at the time was that it would be "unlikely".
As we found out recently for BattlEye though, developers just need to send off a message to have Proton support for Linux and the Steam Deck just turned on which ARK and Mount & Blade II: Bannerlord have done already. Given that, perhaps the situation might be different for games that use BattlEye and so we decided to find out.
When sending over a message to Bohemia mentioning Linux, Proton and the Steam Deck while referencing the new information on how easy it should be to turn on BattlEye support, they replied to say: "Hi there, we're not able to confirm support for it yet, but it's something we're looking into." - which is much more positive than the previous answer.
If we get any more info, we'll let you know.