Perrin originally defined the Avogadro number as the number of atoms in one gram of hydrogen (later it was redefined as the number of atoms in 12 grams of carbon-12). As such, it was certainly discoverable from measurements. However, picking grams, or hydrogen was not "natural". Neither was picking circles and diameters in the definition of $\pi$. Once they were picked the value became discoverable. An additional difference, of course, is that the value of $\pi$ can be established without empirical observations, while the Avogadro number requires them, but that does not seem to matter for invention/discovery.
In November 2018 the 26th General Conference on Weights and Measures voted to change the definition of the mole to be the amount of atoms/molecules given by the Avogadro constant, whose numerical value they fixed by decree (of course, it came from the measurements of the Avogadro number). Since the meter is still the length of the path traveled by light in vacuum during 1/299792458 seconds (originally, one ten-millionth of the distance from the North Pole to the equator) it is currently more "discoverable" than the mole, see History of the Meter.
The mole displaced the concept of equivalent weight (the weight of a given substance which will combine with or displace a given weight of another substance), and the weight ratio is not up to convention. Mole allowed the standardization of these weights by making an arbitrary choice of unit. It is the same with the meter, the ratios of segments are objective, the "unit" segment is not.
In all three cases we have elements of convention intertwined with objective elements, so the choice of word (invention/discovery) is a matter of taste.