Dont worry , one thing ayia napa isnt short of is bars , you have the main square as you know but there are plenty of other bars and tavernas within a spit of the square with singers , kareoke etc
http://www.discoverayianapa.com/section.php?id=1
this will give you an idea , try the queen vic , or if you want a show the heaven rock garden , they do an amazing display with vodka (i think) the bar , a lighter and some one playing firestarter by the prodigy
hope this helps