The hyperfine coupling constant a depends on I,mu,J and the hyperfine field B. For one of the isotopes we know I and mu and we can measure the coupling constant. The hyperfine field B can be calculated for an S electron and depends only on Z and the number of electrons so for example if we look at a hydrogen-like ion we can calculate the hyperfine field easily. If we produce the hydrogen-like ion and measure the coupling constant after sufficient time for the ion to have reached its ground state, we will be able to determine J.
Now for the isotope with the unknown magnetic moment, we produce its hydrogen-like ion and measure the coupling constant again after sufficient time. The J should be the same, the hyperfine field should be the same (does not depend on N, only Z), and the nuclear spin I is known. So from the coupling constant we should be able to extract the magnetic moment of the second isotope.