In the game settings choose a path for your prefix (preferably in your Lutris folder) and select 32-Bit architecture.
Select the TeamSpeak installation file as the executable (you don't have to move it anywhere, just keep it inside your downloads folder).
Under starter options select latest lutris-fshack as the wine version (lutris-fshack-7.2-x86_64 at the time of writing).
Also disable DXVK. This will prevent the SaltyChat window from being just a black screen.
Press save and launch it. Install TeamSpeak as usual. Don't launch it after the installation.
Click on "Wine-Configuration" and select Windows 10 as the windows version. Press "OK".
Go back to the configuration.
Select the TeamSpeak executable instead of the installer as your executable. It's most likely $WINEPREFIX/drive_c/Program Files/TeamSpeak 3 Client/ts3client_win32.exe.
Open the plugin with an archive manager and copy (and merge) the plugins folder into the TS3Client directory. It should be $WINEPREFIX/drive_c/users/$USER/AppData/Roaming/TS3Client/ if you did not change it.
Open a terminal and install dotnet461 with winetricks.