What is the Best Tool to Measure Cartridge Azimuth?

melm, I hate to say it, but for stuff like this, the ears are woefully inaccurate.  To begin with, most of us do not have equal acuity in both ears, not to mention equal frequency response sensing, and not to mention room effects, etc.  And then there's the brain...
However, I also have no beef with just sitting the cartridge squarely at 90 degrees to the LP surface.  This is a recent and radical change of thinking for me.

By the way, I agree that there is no need for a Foz if you have an oscilloscope and know how to use it (which many don’t). You still do need the test LP with 1kHz signal in one channel only. (Needs two bands, one for L channel only and one for R channel only.). Shure made such an LP for the Signet Cartridge Analyzer, for one example. And I agree that in the end it may be best to just adjust the stylus so it sits properly in the groove, regardless of crosstalk, in which case you are relying upon the manufacturer to have built the cartridge to exacting standards, which some do.  99% of the time, I adopt the latter leap of faith.  I think the obsession with azimuth may lead to more alignment errors and more distortion and stylus wear than one would care to admit.

I use a now ancient Signet Cartridge Analyzer along with its companion test LP made by Shure.  The SCA has a ‘scope output, but if you have no scope, it has a dB meter built onto the faceplate.  So, play test LP with (usually) 1kHz test tone in one channel only, set the signal appearing in the driven channel to 0 dB, and measure dB of output in the opposite channel, shown on the meter as a negative number of dB.  Then play the band with a 1kHz test tone only in the opposite channel and likewise measure the output in -dB that appears in the non-driven channel on the meter (or on your scope).  Alter azimuth from 90 degrees to get the least crosstalk, L to R vs R to L.  In my experience, it is folly to try to achieve exactly equal crosstalk levels, R to L and L to R.  Best to just go for the best absolute values.

I would not recommend setting azimuth by using any combination of any bubble levels.   The opportunity for error is too great.  After all is said and done, it is not really a bad idea to set the cartridge visually typically using a mirror so that the cantilever is at a 90 degree angle to the surface of an LP.  For most well made cartridges from reliable manufacturers, this will work fine.  If one is crazed enough to want to set azimuth electrically, you can use a Fozgometer for sure or any of a few other methods for minimizing crosstalk. To do this, you need a suitable test LP that has bands with signal only in one and then only in the other channel.  Based on what you say about your equipment, you probably can figure out how to set for minimal crosstalk.