An update on this issue: it took nearly four weeks, but between Printful and Squarespace communicating to find the problem and then Printful was working with AWS, they did finally find the API issue and fixed it Amazon fixed it (it was on their end -- nothing to do with Amazon.com, but with the AWS servers that are used by Printful I believe). My site is now working as normal and can sync properly with no dreaded Error.
It was good to have an actual product identified as causing the issue so that it could be fixed and all parties worked together to get it done. So I appreciated that all three platforms got involved to find the bug. It has only been a few days, but so far so good.