I actually had the same problem as you and spent quite some time trying to find the reason behind it. My situation was exactly the same:
Launching GameCube games with the Dolphin emulator as standalone and my controller (An 8bitdo Ultimate) worked just fine
Launching GameCube games via Launchbox and my controller did not work.
I think this problem might have started when Launchbox introduced the auto update of emulators via its GUI - but I'm not sure.. Anyway the problem I had (and that you might have is this):
On my computer I have my Dolphin emulator folder - let's say it is: "C:\Emulators\Dolphin-x64\". Whenever I started the dolphin emulator from there everything worked just fine. But, then I realized there was another folder within that folder called "Dolphin" as well. This folder seemed to contain all the necessary files as well but when I ran the "C:\Emulators\Dolphin-x64\Dolphin\Dolphin.exe" file my controller did not work. If I used the "C:\Emulators\Dolphin-x64 \Dolphin.exe" file though, the controller worked just fine.
I checked the Emulator setting for Dolphin i Launchbox and it was pointing to the "C:\Emulators\Dolphin-x64\Dolphin\Dolphin.exe" file instead of the "C:\Emulators\Dolphin-x64\Dolphin.exe" file. Once I changed it, everything worked just fine again.
So, make sure you are pointing to the "correct" Dolphin.exe file from Launchbox via Menu >> Tools >> Manage >> Emulators >> Dolphin, and check the Application Path: