Our DD had to be on two meds at one time for her GERD. The doses had to be increased frequently, as they are dispensed by weight, and even a few ounces in weight can cause the medicine not to be effective. We were also on prevacid, 3 times a day (which is three times the dose they normally give), and reglan. They didn't want to give the reglan, but it was necessary. In addition we were switched to prescription formula because all the meds weren't cutting it. Gastro determined it to be a dairy and soy protein intolerance/allergy.
She didn't have an EGD, but did have an Upper GI Xray - which ended up not requiring sedation - and is not invasive. They gave her a bottle of liquid to drink that appeared bright on the moving xray, so they could follow the path. They also looked at structural stuff and ruled out pyloric stenosis.
She did have general anesthesia during her heart surgery, and as Jamie said, it is never easy. Just make sure it's at a children's hospital, or done with an anesthesiologist who work with children frequently. It'll be worth it to find comfort for her.