I am sorry - my apology for not understanding the issue at hand.
If the system work with both once installed - then I don’t understand the urge to make it work with the live system.
I mean - it is known that the live ISO may not be optimally configured when using the proprietary driver (only Nvidia) is an issue - perhaps (very old AMD as well) - and that you work around by using the free GPU option.
Exactly why I say like I do - you cannot play games on the live ISO - it is not designed for that task - the live ISO is designed for a preview of the desktop and subsequent installation.
That may not be possible unless you create a system snap-shot before applying an update.
moving forward
When your goal is to troubleshoot your system’s GPU drivers, there is a better way.
In the topic Basic Rescue and Recovery, it has been described how you can use either TTY or kernel cmdline to reach a console without the need for a live ISO.
Using one of the described methods (see quotes below), you can reach a running system and perform tasks required.
When troubleshooting a GPU issue I suggest you start with disabling the display manager, usually sddm but there is also the new kid plasmalogin which must be installed by hand if desired, so example is for sddm.
sudo systemctl disable --now sddm
When troubleshooting, having the system booting to a console is - at any measure - to be preferred over booting a live ISO.
You can start the plasma desktop by using one of two
startplasma-waylandstartplasma-x11(only available with plasma-x11-session)
When you are ready - re-enable the display manager (example sddm)
sudo systemctl enable --now sddm