What does the size of the government have to do with breaking up monopolies? Does a multi-billion dollar expenditure on the Department of Education somehow make it easier to break up monopolies?

It needs power to break up monopolies, and a certain minimum size to be able to function effectively in this capacity, yes, but performing this function does not require massive staffs of bureaucrats making tons of regulations on every conceivable industry and thus being susceptible to capture in so many places. These two things are totally orthogonal; you can easily have either of "massive regulation" and "monopoly busting actions" without the other. Minimum size for monopoly busting would not be very large; monopolies aren't that tricky to detect.