Is there a way to prevent the PC/Mac error sound from playing?

Yes.
You can change the PC/Mac sound setting so you won’t hear the error sound.

■ Windows
Go to [Control Panel] > [Sound] > [Program Events] and turn off all sound settings.

■ Mac
Open [System Preferences] > [Sound] panel > [Sound Effects] and change the device to something other than the DJM-V10.