I can't believe I didn't think to do this earlier in the discussion, but I looked up the term in The Dickson Baseball Dictionary, copyright way back in 1989. It has "bat around, v.: when all nine batters in a team's lineup come to bat during an inning. First reference: 1880, Chicago Inter Ocean, May 19." It's just one source. But it's a source, and seemingly researched. I remain solidly in the 9 batters camp.