Mouse button support on the output, not just the input, side

Using latest BTT on latest MacOS. I've long used a couple of all-app mappings, Mouse Button 3 to Cmd+left (on a browser, makes it go back to the previous page) and Mouse Button 4 to Cmd+right (makes it go forward in the history -- back to the future, as it were).

Now, I'm RDPing to a Windows box, and those (sending Cmd-anything) brings up its Start menu. Yes, I know, it thinks Cmd is Windows when physically, Opt is, on my external keyboard... That's a whole other story, and something I'd love BTT to be able to remap on an app by app and maybe even keyboard by keyboard basis, but I digress.

So, my first inclination is to just let it send Mouse Button 3 to RDP when I hit Mouse Button 3. Make it basically a no-op or pass-through. Windows is set up to use those for my intended purpose by default. But... How to tell BTT to send a Mouse Button 3 press? There doesn't seem to be a way to generate anything other than the left/center/right button clicks. That seems a bit inconsistent, no?

Now it looks like I have to take my All Apps shortcuts, and copy them to every app individually... except the one or two where I don't want them. There's not a way to tell BTT not to apply an All Apps shortcut, is there?