The slang phrase “The big cheese” will not trigger lactose intolerance.
“The big cheese” and “the high muckamuck” are roughly synonymous phrases, one, of course, with American origins and referring to food, and the other of foreign heritage, with no specific relation to food. Both slangishly mean the bigwig, the grand poobah, the head honcho, the person in charge.
What’s my point? Everything you know about English is wrong, of course. The American food-related phrase is “high muckamuck,” Anglicizing the Chinook native American phrase hiu mucka-muck, meaning “plenty of food.” Someone who had plenty of food back in the very early 1900s when the phrase was first recorded apparently was pretty well off.
The big cheese, on the other hand, almost certainly comes from a Persian or Urdu word meaning “thing”: chiz. It’s said that in the early 1800s, Anglo-Indians slangishly “translated” the phrase “the real thing” into “the real chiz,” which, by homonymic association, came to be spelled the same way as our curdy mucka-muck. The word wandered across the pond to America, where it came to mean first wealth and then the one with the wealth (and likely plenty of food, too) in the early 1900s.
So any etymology claiming that the original big cheeses were those monstrous cheese wheels is a “Swiss cheese etymology”— full of holes.