I think that the best book of such kind is the monograph by Claus Müller (1969) [1], which is the translation of an older 1957 monograph: first of all, the Author was the mathematical physicist who proved the existence of the solution to the problem of diffraction of electromagnetic waves by a dielectric ball, together with Hermann Weyl and Victor D. Kupradze.
The author does not use distribution theory: by using the integral definition of standard vector operators, he defines a sort of weak derivatives in the following way:
$$
\begin{split}
\nabla\cdot \boldsymbol{a}&=\lim_{G_i\to x}\frac{1}{\Vert G_i\Vert}\int\limits_{\partial{G_i}} \boldsymbol{n}\cdot \boldsymbol{a}\,\mathrm{d}\sigma\\
\nabla\times \boldsymbol{a}&=\lim_{G_i\to x}\frac{1}{\Vert G_i\Vert}\int\limits_{\partial{G_i}} \boldsymbol{n}\times \boldsymbol{a}\,\mathrm{d}\sigma\\
\nabla\varphi&=\lim_{G_i\to x}\frac{1}{\Vert G_i\Vert}\int\limits_{\partial{G_i}} \boldsymbol{n}\varphi\,\mathrm{d}\sigma\\
\end{split}\tag{1}\label{1}
$$
where
- $\boldsymbol{a}$ is a (non-differentiable) vector field in $\mathbb{R}^3$,
- $\{G_n\}$ is an contractible indexed family of smooth sets in $\mathbb{R}^3$ converging to the point $x\in\mathbb{R}^3$, whose volume is $\Vert G_n\Vert$ and whose boundary surface is $\partial{G}_n$,
- $\boldsymbol{n}$ is the inward normal vector to the surface $\partial{G}_n$.
- $\varphi$ is a (non-differentiable) vector field in $\mathbb{R}^3$.
The entire first chapter is devoted to the development of the vector analysis for the "generalized operators" defined by \eqref{1}. After having introduced the standard spherical harmonics and Bessel functions in chapter 2, by using the vector analysis defined in the first chapter, the Author studies Maxwell equations by using time-harmonics waves, i.e by transforming them to properly defined reduced wave (Helmholtz) equations
$$
\begin{split}
\Delta\varphi+\varphi&=0 \text{ (scalar field)}\\
\Delta\boldsymbol{a}+\boldsymbol{a}&=\boldsymbol{0}\text{ (vector field)}
\end{split}\tag{2}\label{2}
$$
Then the study follows by developing the theory of EM waves in homogeneous and inhomogeneous media, and boundary value problems for equations of type \eqref{2}. Finally, you can find more details on its contents by reading the long ZBMATH review of the 1957 edition.
Edit. I noted the OP later EDIT and felt to add the following observation. It is not customary to develop the theory of Maxwell Equations in the time domain, so if you are searching about information on hyperbolic equations in a text on electromagnetism, probably you will not find what you are looking for. It is customary in the electromagnetic theory to study the so called Helmholtz equation which can be deduced from the wave equation by using time-harmonic EM fields, i.e. fields of the form
$$
\boldsymbol{A}(x,t)=\boldsymbol{A}_0(x,\omega)e^{-i\omega t}
$$
The Helmoltz equation is elliptic, and essentially the authors who if you adopt this choice develop the 'Potential theory' of this equation.
[1] Claus Müller (1969)[1957], Foundations of the Mathematical Theory of Electromagnetic Waves, Grundlehren der mathematischen Wissenschaften 155, Springer-Verlag, pp. VIII+353, MR0253638, Zbl 0181.57203.
