Here is a detailed guide for anyone struggling to get multiplayer working:

Avoiding performance issues (frame dropping and stuttering in combat):

1. Install BF3 from the EA app via Lutris (this is the easy part, as you can find tons of tutorials on YouTube)
2. In case you want to download multiple games through the EA app, I recommend you duplicate it inside Lutris to avoid conflicting configurations (e.g. I was unable to run both BF3 and BF4 in the same EA app due to different DirectX installations)
3. In Lutris, select your BF3 EA app and click the arrow up ▴ beside the "Play" button, go to "Configure" -> "Runner options" tab
4. Change the "Wine version" from the default lutris wine to "lutris-GE-Proton7-35-x86_64" (I find Proton GE is way more optimised for both BF3 and BF4)

Updating punkbuster client (avoid getting kicked from multiplayer):

4. Punkbuster needs to be updated for this game, you can just do a quick search for "punkbuster update" on google and download it from the first link (Select "For All Windows Games" version)
5. Extract the downloaded .exe to your BF3 root folder
6. Go back to Lutris, select your BF3 EA app, click the arrow up ▴ beside the Wine button and select "Run EXE inside Wine prefix" (you might need to have "Winetricks" installed for this)
7. Navigate to your BF3 folder and double-click pbsetup.exe
8. After installing punkbuster, you should see BF3 listed. Just click the update button to make sure it's all good

Installing DirectX (fixes Origin launch error message):

9. You have to install the right DirectX version for BF3. You can just go back to Lutris and click "Run EXE inside Wine prefix" again for your BF3 EA app.
10. Navigate to your BF3 folder -> __Installer -> directx -> redist -> double click "DXSETUP.exe"
11. Wait for the installation to finish

Installing a user agent switcher extension to your browser (Battlelog only launches the game from Windows):

12. Download a "User-agent switcher" extension to your default browser where Battlelog should open
13. Open BF3 from the Lutris EA app
14. Change your Battlelog tab/browser user agent to a Windows browser and refresh the page
15. Select your favourite server and have fun :D