I had the exact same issue as you in Apex Legends, and I was able to find a workaround. Basically, nvidia reflex didn’t work with dx11 (DXVK) at all, but forcing the game into dx12 (VKD3D) mode worked without issue.
To enable, you just need to set this in your launch options.
command% -anticheat_settings=SettingsDX12.json
Also, you may need to set the nvapi environmental variable, but I’m pretty sure nvapi is already enabled in proton for Apex Legends.
I recommend using proton experimental as it uses an up-to-date version of dxvk-nvapi
The desktop app is called “EA App” not “EA Play” which is where the confusion is coming from. “EA Play” is the game subscription service that’s like gamepass.