TL;DR: If I were to choose an immutable OS to run on my propriety graphics cards I'd choose an immutable distribution with rolling releases or hardware enablement packages, which tend to do a better job of keeping these graphics libraries up-to-date for new hardware.

I don't have a recommendation but I just learned about immutable Linux OSes from this post. I could see benefits of immutable OS files, but I've been skeptical about package distribution like flatpak and snap, at least in their current state.

Dont get me wrong, the workflow of flatpack is great, but in my experience, apps from flatpack typically ship with their own runtime and don't rely on system runtimes (likely why you have GPU driver issues). As a software developer, I obviously prefer to ship with all dependencies and runtimes so I don't have to rely on the system to be updated but this comes with downsides:

A major problem with alternate runtimes is drivers. New graphics hardware needs new graphics libraries which have a ton of dependencies. Mesa depends on LLVM for compiling shaders. The NVidia driver depends on a kernel module whose version must exactly match that of the library. All of these libraries have their own transitive dependencies like libdrm, libstdc++ and glibc. If you want new hardware to work, you need to be using new versions of all of these libraries.

Linux distributions, especially those with rolling releases or hardware enablement packages, do a great job of keeping these libraries up-to-date for new hardware. Bundled runtimes do not. Source.

I'd recommend checking out that article I linked as source. There are also security concerns of using apps, some of which are mitigated by having an immutable filesystem, but there are more points and this comment is long enough as it is.

EDIT: I reread my comment and it comes off as "immutable bad, blah blah". Truth is I don't know much about these OSes but I wanted to point out that distributing apps in containers comes with its own challenges; which I gather is necessary for immutable OSes. So my TL;DR is to narrow down to a distribution that is immutable and has a rolling release or distributes hardware enablement packages.