Tried this myself and had to jump through a couple of hoops to get this running.
I'm using wine 2.4 und winetricks 20170316 on arch here.
Here are my steps. I've used a fresh wine prefix so this should work without any additional steps given that you have wine and winetricks on your system already.
Things to note:
- setup_battlezone98_redux_220.127.116.11.exe will fail to install vc runtime. So we install that before running the setup and thus can ignore the error.
- ie7 is needed for displaying the launcher properly
- directx is needed and not supplied by the setup. also d3dcompiler is needed, otherwise shaders won't compile and screen will stay black.
# setup new wine prefix
WINEARCH=win32 winetricks prefix=battlezone
# install requirements
winetricks prefix=battlezone vcrun2012 vcrun2013 ie7 d3dx9_43 d3dcompiler_43
# install battlezone
WINEPREFIX="$HOME/.local/share/wineprefixes/battlezone" wine ./setup_battlezone98_redux_18.104.22.168.exe
# run it
cd "YourInstallPathHere/Battlezone 98 Redux/Launcher/"
WINEPREFIX="$HOME/.local/share/wineprefixes/battlezone" wine ./BZLauncher.exe