Maxwell's equations do follow from the laws of electricity combined with the principles of special relativity. But this fact does not imply that the magnetic field at a given point is less real than the electric field. Quite on the contrary, relativity implies that these two fields have to be equally real.
When the principles of special relativity are imposed, the electric field $\vec{E}$ has to be incorporated into an object that transforms in a well-defined way under the Lorentz transformations - i.e. when the velocity of the observer is changed. Because there exists no "scalar electric force", and for other technical reasons I don't want to explain, $\vec{E}$ can't be a part of a 4-vector in the spacetime, $V_{\mu}$.
Instead, it must be the components $F_{0i}$ of an antisymmetric tensor with two indices,
$$F_{\mu\nu}=-F_{\nu\mu}$$
Such objects, generally known as tensors, know how to behave under the Lorentz transformations - when the space and time are rotated into each other as relativity makes mandatory.
The indices $\mu,\nu$ take values $0,1,2,3$ i.e. $t,x,y,z$. Because of the antisymmetry above, there are 6 inequivalent components of the tensor - the values of $\mu\nu$ can be
$$01,02,03;23,31,12.$$
The first three combinations correspond to the three components of the electric field $\vec{E}$ while the last three combinations carry the information about the magnetic field $\vec{B}$.
When I was 10, I also thought that the magnetic field could have been just some artifact of the electric field but it can't be so. Instead, the electric and magnetic fields at each point are completely independent of each other. Nevertheless, the Lorentz symmetry can transform them into each other and both of them are needed for their friend to be able to transform into something in a different inertial system, so that the symmetry under the change of the inertial system isn't lost.
If you only start with the $E_z$ electric field, the component $F_{03}$ is nonzero. However, when you boost the system in the $x$-direction, you mix the time coordinate $0$ with the spatial $x$-coordinate $1$. Consequently, a part of the $F_{03}$ field is transformed into the component $F_{13}$ which is interpreted as the magnetic field $B_y$, up to a sign.
Alternatively, one may describe the electricity by the electric potential $\phi$. However, the energy density from the charge density $\rho=j_0$ has to be a tensor with two time-like indices, $T_{00}$, so $\phi$ itself must carry a time-like index, too. It must be that $\phi=A_0$ for some 4-vector $A$. This whole 4-vector must exist by relativity, including the spatial components $\vec{A}$, and a new field $\vec{B}$ may be calculated as the curl of $\vec{A}$ while $\vec{E}=-\nabla\phi-\partial \vec{A}/\partial t$.
You apparently wanted to prove the absence of the magnetic monopoles by proving the absence of the magnetic field itself. Well, apologies for having interrupted your research plan: it can't work. Magnets are damn real. And if you're interested, the existence of magnetic monopoles is inevitable in any consistent theory of quantum gravity. In particular, two poles of a dumbbell-shaped magnet may collapse into a pair of black holes which will inevitably possess the (opposite) magnetic monopole charges. The lightest possible (Planck mass) black holes with magnetic monopole charges will be "proofs of concept" heavy elementary particles with magnetic charges - however, lighter particles with the same charges may sometimes exist, too.
My recent article might be relevant: http://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-013-2371-4.pdf (published in the European Physical Journal C, open access). An excerpt from the abstract: "after introduction of a complex four-potential of electromagnetic field, which generates the same electromagnetic fields as the initial real four-potential, the spinor field is algebraically eliminated from the equations of spinor electrodynamics. It is proven that the resulting equations for electromagnetic field describe independent evolution of the latter"
The article may be relevant, although I do not use the formalism of differential forms, as you "take the Dirac wave function not to be observable in the classical Maxwell-[Dirac] theory" and "prefer to disavow unobservable fields unless there are absolutely clear reasons why they are unavoidable."
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For what it's worth, throughout my career as a professor of theoretical physics specializing in high energy physics and the foundations of Lorentz covariant quantum theory, I never heard of Harmuth or his modification of Maxwell's equations until reading this question and the indicated foreword to his book. Now it's hard to keep track of everything going on in science, even within ones specialty. But I think it's safe to say that Harmuth's theory is not accepted by the physics community as an improvement on Maxwell's equations. There have been proposed modifications that have received extensive attention and study, such as Born-Infeld electrodynamics, and ever since the development of the Electro-Weak unified theory our understanding of the status of the EM field has been altered. But no proposals for modifying Maxwell's equations, per se, have been accepted as established improvements.