If you need to level shift the 3.3V outputs to 5V you could use the techniques described here
The Arduino Leonardo and Pro Micro both have a hardware serial port that you can use (Serial1) because the internal serial to the PC is handled directly inside the microcontroller IC as the IC use don those boards has a built in USB connection, which leaves the normal serial lines free for other uses.
I've bought several Pro Micro's from eBay sellers for around $10.
But I also have a Mega2560 etc
Was there a specific reason you are using the Due, it seems overly powerful for simple midi processing.
BTW. I have a Due on order but it seems to have been lost in the post ;-(