There's a 2021 paper by Carlos P. Roca and his collaborators that describes an easier way to calculate the spillover matrix. I'm mad at myself for not looking into it earlier - AutoSpill, as it turns out, is pretty awesome.
See the Roca et al paper here: AutoSpill is a principled framework that simplifies the analysis of multichromatic flow cytometry data (rdcu.be/c0ONd)
See the UoC CAT Facility Blogpost in which Laura describes the problems that AutoSpill is now fixing: The Right and Wrong Way to Set Up Automated Compensation Tools: How to Achieve Accurate Compensation (voices.uchicago.edu/ucflow/20...)