What you are describing goes well with esophageal irritation associated with acid reflux. Is your daughter otherwise well and thriving (gaining adequate weight, height, and head circumference on the growth charts)? Does she have any respiratory symptoms? (persistent cough or wheezing)? These symptoms go with gastrointestinal reflux disease which may lead to significant esophageal inflammation and thus pain with burping.
The fact that is relieved when put in a horizontal position, however, goes against this explanation: it goes more with colics relieved by expelling the excess gas.