ATMega328P Programs at 2V but not at 1.8V
I visited a friend who has the AVR ICE programmer, and it worked just fine with my board.  My AVR ISP mkII might (?) be counterfeit or otherwise not properly buffered, though it works above 2V.

For anyone else that runs into this kind of issue (programming cutting out at 2V exactly) it's probably your programmer, and it's worth buying the Atmel ICE through a reputable source.

Thanks to @jonk and @Ignacio for the help!