Meaning of "Friday face" in 1592
Solution 1:
A dictionary from 1793 gives the answer:
Friday face, a dismal countenance, before and even long after the reformation. Friday was a day of abstinence or jour maigre.
Blackguardiana
(Note that your source clarifies that it is only “counterfeiting sorrow”.)
For some more context, certain Christian groups (eg Catholics) were prohibited from eating meat on Fridays. Some Christians still observe this (even if only during Lent). See Are Meatless Fridays Still a Thing? Does it Matter?
If I were translating for the Modern English reader, I would use the expression “Monday face” or “☹️”.