I'm trying to write to the part in single-byte mode, but I can't write anything. What am I doing wrong?

Remember that a communication cycle consists of 16-bits of instruction and 8-bits of data (data byte). Make certain that when you finish the 1st data byte, you bring the CS-bar pin high, and then low before attempting to write the 2nd data byte (in another communication cycle). If you do not toggle the CS-bar pin, the serial port controller will flush any new writes, and only the very last write (which is the update itself) will be recorded.