I found that if I install the Windows version of Punkbuster and install punkbuster for BF4 from the standalone pb app (pbsetup.exe), like @Rakanoza said, and also change my wine version to ‘tkg-protonified-3.21-x86-64’ instead of the latest version of wine (4.9), I have a ping in-game. In the server browser, I still get the same ping next to any server, but in-game it’s the correct value. So no more getting kicked with this wine version.
I did notice a slight performance drop with the older version of wine: every 1-5 seconds, I get a stutter that can last up to about a second. I didn’t get this with the latest version of wine.
EDIT: turns out the stuttering was because of thermal throttling. Fixed by lowering maximum CPU frequency.
If I use the latest version of wine + installing standalone punkbuster, I don’t get kicked by punkbuster anymore but a lot of server admins don’t like me having no ping value in-game, kicking me after a minute or two.
What wine version are you using?