Hi,
I am unable to offer anything concrete about William... I'm sorry.
In the reply you received from the Mayflower Society it should have clearly layed out what was known and what was missing. They generally have lines that they have accepted, then your task is to make the proofs from yourself up to where you connect to the line. It sounds like you know right where the issue is, that's a big plus. They likely want you to find documentation of William/s birth showing his parents. For ancestors that far back I understand they sometimes accept less direct evidence such as a mention of his parentage in a publication.
WikiTree is a great place to collect and post all of your known information and to work on filling in the gaps, especially because so many people work together.
I found new information I was unaware of previously by doing just that. I met a cousin who was working some of the same family and found another notable ancestor for me (coincidently another mayflower pasenger). Now we work together all of the time and exchange greetings at holidays.
In the end, only the Mayflower Society can tell you exactly what they want from you.
Have a super day!
