Update: After posting this question, I discovered (well, nobody's perfect!) I had a book in my possession that has addressed this question in some detail, though apparently not found a definitive answer. David C. Cissell wrote "The Pedigree of William Cecil, Lord Burghley" in 2020. It includes several chapters of interest. I am going to write the author and ask for permission to post some quotes (somewhere on WikiTree). If anyone cares about this question, send me a PM and I'll update you.