Can the esophagus be so inflamed it's dangerous to get a biopsy? how else can you rule out cancer?

This is extremely unlikely. It's far more dangerous to not do a biopsy if recommended either to confirm inflammation (esophagitis) or to rule out cancer. If you have seen a gastroenterologist, you can trust her advice on this. Good luck.

Biopsy is always required for diagnosing esophageal cancer, no other alternative method. Yes, during EGD an inflamed esophagus can/should be biopsied if suspected lesion is observed for ruling out malignant disease.