The main problem with your argument is the assumption that all mass originates from interactions with the Higgs boson. This is not the case. For example in a proton the Higgs mechanism is responsible for the masses of the three valence quarks, but these make up only about 1% of the total proton mass. The other 99% comes from the quark interaction energies and this mass would remain even if the Higgs interaction was zero and the quarks were massless. So you cannot use your argument to prove that a singularity has no mass.
But even leaving this aside the concept of mass for a black hole is more subtle than you might think. The archetypal black hole, the Schwarzschild metric, is a vacuum solution i.e. the mass density is zero everywhere except at the singularity where it is undefined. A Schwarzschild black hole does not have a point mass at the centre. In fact technically it does not have a centre at all since we traditionally consider the singularity not to be part of the spacetime manifold. The mass $M$ that we use in the description of the Schwarzschild black hole is a property called the ADM mass that is actually a geometrical property.
Part of the problem is that while we glibly refer to the Schwarzschild geometry as a black hole, it is actually an asymptotic limit. The Schwarzschild geometry is time independent meaning it has existed for an infinite time and will continue to exist for an infinite time into the future. This is obviously not physically realistic.
A real black hole would be described by something like the Oppenheimer-Snyder metric, though this is a highly simplified model. The OS metric forms a Schwarzschild geometry in a finite proper time (i.e. time measured by an infalling observer) and right up until the singularity its mass is just the mass of the ball of dust that is collapsing. So it is a mass in the sense we all intuitively understand.
However the formulation of the singularity isn't described by the OS metric. The metric describes the evolution up to an instant before the singularity forms, but no further. Describing what happens in the final moment will require a theory of quantum gravity that doesn't exist yet. Until we have such a theory we simply can't say what happens to the mass when a black hole forms.