Ah, you seem to be assuming that the mass of the crown is known and an equivalent mass of gold can be produced.
> What we're actually trying to check is that the readout doesn't change ...
I'm not sure what you mean by readout. If one side is A and the other B the balance only has three readings, A>B or A=B or A<B.
Typically I would expect the crown to be put on one side of a balance. Then the weigher would search for the amount of gold that makes the balance, well, balance (A=B). In effect the weigher chooses diff_air==0.
> A good counterfeiter will ensure diff_air==0
I don't see how the counterfeiter has any influence on diff_air, since the mass is not known beforehand.
Your equations seem correct.