The experimental procedure would be quite simple, the hyperfine coupling constant a would be measured for both isotopes. From the video, we know that
a= mu B_J / (I J).
With a_1, a_2 measured and I_1, I_2 and mu_1 given, we only need expressions for B_J and J.
I will make the assumption that J and B_J which only depend on the electron cloud will not change if we add neutrons to the nucleus (which is the only difference in isotopes). (J_1=J_2, B_J1=B_J2)
This means that having measured a_1 and a_2, a simple calculation will lead to mu_2: