Well... this is depressing.
I changed the PCM chip for a new one straight from TI. I was very careful not to burn it, lowering my iron temperature and taking breaks between pins to let the chip cool down.
Plugging the DAC in, it's recognized, yay!
[ 218.102222] usb 5-1: new full-speed USB device number 3 using uhci_hcd
[ 218.284426] usb 5-1: New USB device found, idVendor=08bb, idProduct=2707
[ 218.284439] usb 5-1: New USB device strings: Mfr=1, Product=2,...