It could certainly be reflux, which can manifest itself in multiple ways, including coughing for example. Look up LPR, which is a type of reflux.
Regarding abdominal exercise, I was told not to do sit ups. Instead of raising your head you raise and lower your legs.
You may have a hiatal hernia.
So, don't do what causes you the pain, because aside from the pain it may be causing irritation in your esophagus which you don't feel (silent reflux, but if severe it can cause cancer in the long run if not addressed).
Look up suggestions related to reflux and see if it helps you. And no big meals, no bubbly drinks. May need to stop/reduce caffeine and chocolate, which loosen the sphincter. Last meal a few hours before bedtime. Stretch and exercise...