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Submarines generally have neutral bouyancy while underway so there's not much density difference to discern.

That sounds absurd, but the $5 pressure sensor on my weather station could detect a volcanic explosion 5,000 miles away.

Is there even a difference between a submarine and the surrounding water's gravitational fields when the sub has neutral buoyancy?

There probaly would be, the displacement might be the same but the distribution of mass wouldn't. But I recon these instruments are only sensitive enough to measure the difference between the seafloor and the water above.

I'd argue that the gravity field of a sub is not uniform despite having neutral buoyancy. No idea though how significant that is vs. how sensible the measurement is

You'd have better luck trying to measure a neutron with a bathroom scale.

Not even close.