Avrdude fails at specific baudrates
If I change it back to 115200 baud, it uploads fine...even though the sketch itself initializes the serial port to 57600.


The bootloader is a piece of code designed to work at a specific baud rate. As KIIV said in his answer only the USB-based Arduinos would be relatively immune to changing the rate.

What your own code does, after being uploaded, is completely irrelevant.


  Avrdude fails at specific baudrates


Your title really should be: "Avrdude only works at one baudrate (the one it was designed to use for a specific board)"