Is it possible to mute a specific sound after creating a loop with the X-PAD’s overdub function?

Yes.
When using the overdub function, press the [X-PAD FX] button for the sound you want to mute while pressing the [HOLD (MUTE)] button to mute that sound. When the [X-PAD FX] button for the sound that is muted is pressed again while pressing the [HOLD (MUTE)] button, muting is canceled.