If we can all agree that there is a force/ acceleration, apparent or not, acting 'downwards' then we can all agree that on both a flat Earth and a globe the atmospheric pressure will be exponentially decreasing. The
barometric formula developed by Boltzman, with its foundation in statistical mechanics, makes no assumption on the shape of the Earth. All that is assumed is that there is an acceleration that is constant in height (a fair one for the globe theory). The exponential decrease in pressure is purely a consequence of the statistical thermodynamics and says nothing on the shape of the Earth or whether or not there is gravity.
Just to be more explicit. Here is the important line in the Wikipedia article:
Assuming constant temperature, molar mass, and gravitational acceleration, we get the barometric formula...
So, for all theories there should be a vacuum at very high altitudes.