EAC has a check box for Proton compatibility. Battleeye is linux native. All they have to do is check a box, and test to see if they can break it. If they let it out in the works and there's some influx of cheaters, they can check the box again. Halo Infinite, Apex Legends, Smite, Battlebit etc etc were all capable of checking the box and testing.

I suspect Sweenys hesitation over support is caused by a lack of control.

Upgrading EAC in an unreal engine game is trivial, it's basically baked into the engine. They update EAC all the time.