The Greek Septuagint (LXX) translates this passage from the Hebrew as seventy times seven (rather than 77). The reason that I bring this up is that this translation captures how Greek-speaking Hebrew scholars roughly around 250 BCE interpreted this number.
Regarding Base 60 (sexagesimal), there's no affinity of this mathematical system to the number 7 (the Sumerians passed Base 60 down to the Babylonians). It is indeed a more convenient base for trading fractions of quantities than Base 10. Base 10 has 2 pairs of divisors, Base 12 has 3 pairs of divisors, and Base 60 has 6 pairs.
One could argue that Base 8 (octal) would be a more likely candidate than 60 since God created the heavens and the earth in seven days. Day eight in Base 8 would then be written as 10 in that case. In other words, one would count 1,2,3,4,5,6,7,10, 11, etc. In Base 8, 70 would be 56 in Base 10 and 70 x 7 in Base 8 would equal 392. However, no convincing evidence has been found of the use of Base 8 in the Bible or anywhere in the ANE. Octal was used by the Yuki tribe in California and the Pame people in Mexico.
Calendars were absolutely crucial in determining when to plant crops. The lunar calendar was used with 12 months of 30 days to yield a 360-day year, which does have some affinity to Base 60. The extra 5.24 days in the actual 365.24-day year were not counted (I think they were considered holidays). The average lunar month is close to 29.5 days, but the number of weeks in a month is only 4, so a four-week month is only 28 days. This does not work out evenly. Since the new moon is pretty obvious, ancient peoples would be aware of at least a one-day discrepancy per month, which would need to be accommodated when keeping the Sabbath in a seven-day week.
Also note that various ancient civilizations had different numbers of days in a week, but the ancient Babylonians used seven days as did the Hebrews and other peoples in the ANE.
I really can't see any way that a different base could impact P.J. Wiseman's Tablet hypothesis regarding Genesis.