First of all, I would use my experimental set up to calculate the coupling parameter a of the first isotope with I1 and μ1 known.
After that, we will be available of building the energy diagram for the isotope 1.
If we apply the same magnetic field to the isotope 2, the coupling parameter will be the same, but the energy levels will change due to the change in the nuclear spin moment I2. Measuring the energy lines between those new energy levels, and using the coupling parameter calculated previously, we will be able to calculate the magnetic moment μ2.
The electronic part J doesn’t change between two isotopes. Then, we can write the relation
In our first isotope, we can obtain a1 empirically, and we know I1 as will as mu1.
In our second isotope, we can also calculate a2 empirically, and we know that J will remain the same as in isotope 1, meanhile mu2 and I2 will change. We know I2 value, so if we apply the same magnetic field in both experiments, relating the two expressions we can extract mu2 as a function of a1, a2, I1 and I2, all known values.