The Nintendo Switch Pro Controller should work, but LaunchBox itself does not handle controller input once a game is launched. LaunchBox hands the game off to RetroArch, so in-game button recognition is controlled by RetroArch and Android.
First, please confirm which RetroArch app LaunchBox is using for your Sega Genesis platform: RetroArch, RetroArch Plus, or a sideloaded RetroArch build. Each one has its own separate controller/config settings.
Try opening that exact RetroArch app directly, launch the same Genesis game manually, then go to:
Settings > Input > Port 1 Controls > Device Index
Make sure the Nintendo Pro Controller is selected for Port 1. Then check:
Settings > Input > RetroPad Binds > Port 1 Controls
and bind the buttons there. After that, save the RetroArch config and launch the game again from LaunchBox.
If the controller does not work in RetroArch directly either, then the issue is with Android/RetroArch seeing the controller, not LaunchBox. If it works directly in RetroArch but not when launched from LaunchBox, let us know which RetroArch version/package LaunchBox is set to use and which Genesis core you selected.