In my college days, almost 50 years ago, we did look at the sodium doublet with a Michelson interferometer, which works on similar principles as the Fabry-Perot interferometer. Perhaps the biggest difference is the higher reflectivity of the Fabry-Perot, where the Michelson has close to 50% reflectivity, and thereby the bright rings from a spectral line are broad, rather than very narrow as with the high reflectivity ##R ## of the Fabry-Perot.
The formula that is used for ## T=T(\lambda, \theta) ## with the Fabry-Perot ( see the wiki link above) reminds me very much of the diffraction grating type spectrometer where we use a formula for ## I=I(\lambda, \theta ) ## in order to compute the resolution by determining the observed linewidth of a laser type spectral line. With the diffraction grating spectrometer we have ## I=I_o \frac{\sin^2(N \phi/2)}{\sin^2(\phi/2) } ## where ## \phi=(2 \pi/ \lambda) d \sin{\theta} ##. It's a very similar calculation like that of above to show resolving power ## \lambda/\Delta \lambda=N m ## for the diffraction grating spectrometer.
Relevant to the OP's problem of observing the Zeeman effect in hydrogen, I did observe the Zeeman splitting of the green line in mercury ( Hg) in a magnetic field into 9 components if I remember correctly, with a diffraction grating spectrometer in an upper level undergraduate laboratory spectroscopy course for which I was a teaching assistant.
I should mention in all cases, whether it is hydrogen, sodium, or mercury, it is a gas discharge lamp that is used to observe the spectral lines. For the Zeeman effect, the magnetic field is applied normally with a ring shaped permanent magnet with a gap about one inch across between the north and south poles. The arc lamp is positioned with the part that is being observed in between the pole faces of the magnet.