I did not know BTT supported this until today!
You can just set up the trackpad 4-finger swipe actions globally: https://cleanshot.com/share/P0K1PGC1
Then in System Settings set "swipe between full-screen applications" to "off" in Trackpad settings under "more gestures" so that BTT's shortcut applies instead of the system-level one.
Works well. No extra software needed if you already have BTT, which is worth the money for me purely for "alt+drag a window from anywhere" style window movement. That setting is buried deep under BetterTouchTool Settings → Window Snapping & Moving → Moving & Resizing Modifier Keys: https://cleanshot.com/share/mnF9xBkW