Way to use other baud rates in console or different input/output rates
The Arduino Uno is powered by an ATmega328P microcontroller. This
microcontroller has only one UART which is not capable of using
different baud rates for sending and receiving.

You could look into the different software serial port implementations
for the sending side. Another solution would be to continually change
the baud rate: each time you get a complete message from the serial
monitor,


you switch the baud rate to 31250 bps
you send the packet to the midi synth
you Serial.flush() to make sure the bytes have actually been sent
you switch back the baud rate to whatever you used in the serial
monitor


You will probably see some garbage on serial monitor as it receives the
bytes at the wrong rate, but that should not be an issue.