As far as I know, there is no way to use the MUX switch on Framework 16 on Linux. Framework 16 does not have a MUX switch toggle in the BIOS and relies solely on automatic MUX switching using AMD smart access graphic technology, which currently only works on Windows. On Linux, FW16 should behave like a MUXless laptop. So the laptop screen is connected to the iGPU on Linux, and without a working MUX this cannot be changed. Expansion Cards are always connected to the iGPU and a working MUX would not change that. USB C on the back of the laptop is always connected to the dGPU. I don’t think your problem is caused by the MUX, because MUX switching doesn’t happen on Linux. If we had a MUX switch toggle in the BIOS you could select only one gpu and that might help solve your problems, but unfortunately that is not possible. You might find some useful information in this thread.
OpenGL applications should run on the iGPU unless they are started using DRI_PRIME=1 or are configured use an alternate gpu in their .desktop files. Vulkan apps can automatically choose a gpu. You could probably force them to use the iGPU if you started them using DRI_PRIME=0!. (The ! character is part of the command.)
Maybe you could somehow completely disable the dGPU in the operating system. Some people have done this on INTEL+NVIDIA laptops. I’ve never done it so I’m not able to tell you the exact steps.