Intel 13th generation DRAM issues reported
Intel 13th generation DRAM issues reported
Yea, that's the risk with building Intel 13th with the previous 12th gen chipsets as it's highly likely you'll be getting older stock on pre-13th Gen supported BIOSes. You could go to a local computer shop that offers system builds to see if they can do the BIOS update for you with a 12th Gen chip they might have laying around. Be prepared to get charged maybe 20 bucks or so as a service fee if they won't do it for free (nothing against that, you're not entitled to free service). Either that or return the B660 board and get a B760 board instead. Doing a quick search and the prices honestly aren't that much different.
The BIOS on any motherboard isn't certain. A store can't confirm it unless they open the case and test the board. You'll probably face a similar issue. B760 boards should generally function with 13th generation chips right away.