A few users reported they couldn't switch weapons after they received them, had a bit of a play myself to find out what's going on.
You are running 2 instances of the mod! The gzdoom.ini included in the mod is already configured to autoload the GBA Doom II TC.
If you drag the WAD on top of GZDoom, it'll be loading the TC through the ini and also as a command line (there will also be warnings talking about duplicate entries on launch like in the above pic)
To fix it, just launch GZDoom.exe on it's own.