Feedback from Autodesk was that the ability to analytically invert a tone mapping function has a lot of important use cases through the industry. It turns out my tone mapper was quite difficult to invert, but a very small and almost imperceptible change made it trivial to invert. The key is making the desaturation step not change the compressed brightness value, so instead of mixing toward pure white [1, 1, 1], instead go toward peak * [1, 1, 1]. Anywhere the mix is significant, peak is already very close to 1, so this is a minor change, but it helps enormously with invertibility since these asymptotes become so sensitive when inverted.

I've included an analytical inverse function in lut-writer.mjs and verified the round-trip relative error is less than 2e-10.