Got !FS working in the browser using PyScript! It's pretty cool - python, skyfield, numpy etc running inside the browser, fully client-side. And I didn't have to modify the code, it just works. And most importantly, it performed pretty decently. Acceptable performance.
The performance on desktop browsers is pretty good. It's a bit slower on mobile (but acceptable for my purpose).
sgp4 has some C-bindings, so I had to compile to WebAssembly using Emscripten, and made a .whl (wheel). skyfield is pure python, so I made a wheel for that as well using python -m build.
It all works surprisingly well.