Sources in Born-Infeld Electrodynamics What are the ways that charge/current densities are incorporated into the Born-Infeld Lagrangian?
In the paper Point Charge in the Born-Infeld Electrodynamics it appears that 4-current is incorporated by simple addition (though it's not explicitly given there):
$$L = -b^2 \sqrt {1-\frac {(E^2-B^2)}{b^2} - \frac {(\vec E \cdot \vec B)^2}{b^4} } +b^2 + j^{\mu} A_\mu.$$
However, it seems to me that it would make a bit more sense to incorporate it this way:
$$L = -b^2 \sqrt {1-\frac {(E^2-B^2)}{b^2} - \frac {(\vec E \cdot \vec B)^2}{b^4} + j^{\mu} A_\mu} +b^2 .$$
Has this  alternative been explored, and are there other alternatives that have been explored?
 A: If the theory has point-like sources then inclusion of a current under the square root is problematic. The current field of a point charge $q$ has a delta-like singularity:
$$
j^0(t,\mathbf{x}) = q \delta^{(3)}(\mathbf{x}-\mathbf{x}_q(t)),\qquad \mathbf{j}(t,\mathbf{x}) = q\, \mathbf{v}_q(t) \delta^{(3)}(\mathbf{x}-\mathbf{x}_q(t)).
$$
If the coupling term $j^\mu A_\mu$ enters the Lagrangian linearly, then its integration over spacetime would produce a well-defined quantity. However if we insert this term inside a nonlinear function like a square root then the result would be undefined: Taylor expansion of the square root would contain powers of delta-functions.
If the theory only has charged fields rather than point charges then it is indeed possible to include coupling term under the square root but only in combination that ensure the gauge invariance of the dynamics. For example, for complex scalar field $\Phi$ one such combination would be $(D_\mu \Phi) (D^\mu \Phi)^\dagger$, where the covariant derivative is defined by $D_\mu\equiv \partial _\mu -i q A_\mu$. This expression also contains the kinetic term for the scalar field. Whether to include any additional terms under the square root depends on the origin of the theory and goals one tries to accomplish with it.
Today, the relevance of Born–Infeld theory largely comes from string theory where it arises as low energy effective action of D-branes, and so string theory offers heuristics about how one can incorporate specific features for Born–Infeld-like field theories. For more information on Born–Infeld action in string theory I would recommend reviews by Tseytlin and by Schwarz.

For example, Dirac–Born–Infeld action of D$p$-brane ($p$ is a number of brane's spatial dimensions) in a static gauge is
$$S_\text{DBI}=T_p\int d^{p+1}σ\sqrt{−\mathop{\rm det}(η_{αβ}+∂_αX^i∂_βX_i+ 2πα^′F_{αβ}) }.$$
Here $T_p$ is a brane tension, $X^i$ are transverse coordinates and a critical field strength of the Maxwell field is inversely proporional to string constant $α^′$. The $\mathop{\rm det}(η + F)$ structure is a generalization of expression under square root valid for arbitrary dimensions. When $p=3$ and $X\equiv 0$ one recovers an ordinary Born–Infeld Lagrangian.
From the worldvolume point of view  $X$'s are scalars, and they do not carry a charge. But one can consider a non-Abelian generalization of DBI action where both $F$-field and $X$ coordinates become non-Abelian/matrix-valued. In string theory such enhancement occurs for a stack of several coinciding D-branes. If this enhanced gauge symmetry is subsequently broken then, depending on the details, we could end up with a field theory containing at low energies an (Abelian) Maxwell field and one or several complex scalars now carrying charges with the action of the form
$$
S=T_p\int d^{p+1}σ\sqrt{−\mathop{\rm det}(η_{αβ}+(D_α\Phi) (D_β\Phi)^\dagger+ b^{-1} \, F_{αβ}) }+V(|\Phi|)+…,
$$
where $V$ is the potential (possibly with mass term) for the scalar field and by dots we denote all the terms irrelevant to the dynamics of scalar and Maxwell fields. The lowest order terms of Taylor expansion for this action near $\Phi=0$ and $F=0$ would give an ordinary scalar electrodynamics. 
