I definitely don't think you were BU for blowing up at your sister! In fact, not only would I have blown up, I would have left! IMO, it was selfish and thoughtless, not to mention unfair to the child!
But wait... where were the parents? Are sister and her DH raising this GC? If not, why couldn't one of the parents stay home w/ the child? Why palm GC off on your sister and BIL? If the parents couldn't stay home, did they know that sis and BIL had plans I wonder? Would they be ok w/ their taking their sick child out when they weren't feeling well? I imagine not! Sister was put in a difficult position, of course, but IMO, she should have rung/texted/emailed you to cancel or postpone the plans.