The biggest problem is anti cheat software. As soon as they figure that out it’ll get a lot higher!
Obviously, the installer tries to install and start PB then errors out quite spectacularly, as one might imagine such a program to do in a wine prefix, but then you can just download the PB executable and run it in the wine prefix post-install no problem. Origin plays surprisingly nicely, too, though it generates a total of six windows that you can't close during play or it freaks out. There is some additional fuckery required; Wine's networking needs a bit of massaging to allow BF4 to advertise its ping to multiplayer servers, and you'll get kicked if you've got a ping of "-". [0]
Thing is, though, Proton is Wine-and-allied-trades. In the fullness of time I suspect new BF4 players won't have to jump through these hoops as Proton generally, and its script for BF4 specifically, gets updated. And others are already racing ahead, too, borrowing from and providing for Valve's fork of it. GloriousEggroll, recommended for BF4 [1], is the most robust varietal currently.
Multiplayer in a general sense is going to be a little bit more difficult to enable than merely waiting for updates, IMO. I'm not savvy enough to properly understand the arguments, but I've read that the translation layers for graphics, DXVK et al, could easily be repurposed by clever enough end-users to eg wallhack by making textures transparent, etc
As sibling comments point out, resistance on the part of the devs (or publishers?) to simply enable the Linux support that already exists is probably the biggest hurdle.