I could not just approach my employer and ask if I could help code this open source project.

Why not? Every open source contributor I know is because we worked together and they got paid to write open source software that was valuable to the company.

I know many do it outside of work and my sample set is certainly biased, but I also know that a lot of companies will pay for it if they use that software and see value in it getting better.

Certainly doesn't hurt to ask!