If you are not comfortable fishing more than one rod then it is best, as others have stated, to stick to using just the one.
Once you feel more proficient then fishing your normal method with a second "sleeper" rod set-up on an alarm is perfectly fine and easy to do.
That said, take a look at Carp anglers all over the Country who regularly fish at the same time which is obviously best done with bite alarms and bait-runner reels. Or indeed in France where we use 4 rods.
In my experience I've only ever had 4 "double headers" so you simply play and land the first having set the hook on the second, engage the bait-runner mechanism and allow that fish to roam.
On all 4 occasions I landed both fish perfectly well with no obvious signs of distress of any kind, to the fish that is.