We can wlog consider a reference frame where the plasma medium is static, i.e. $V^{\mu}$ has only a temporal component:
$$-1 = ||V||^2~=~g_{00} V^0V^0 \qquad \Rightarrow \qquad V^0 ~=~ (-g_{00})^{-1/2}. \tag{A}$$
We assume that the metric $g_{\mu\nu}$ does not have any mixed temporal-spatial components.
Eq. (3) in Ref. 1 goes to the center of the Abraham-Minkowski controversy$^1$. Eq. (3) is equivalent to Minkowski's proposal that the photon 3-wavevector in a refractive medium is
$$|{\bf k}|~=~n\frac{\omega}{c},\tag{1.10}$$
cf. eq. (1.10) in Ref. 2. Equivalently$^2$
$$ n^2~\stackrel{(1.10)}{=}~\frac{g_{ab}k^ak^b}{-g_{00}(k^0)^2}
~\stackrel{p=\hbar k}{=}~1+\frac{p^2}{(p^0\sqrt{-g_{00}})^2} ,\qquad
p^2~:=~ g_{\mu\nu}p^{\mu}p^{\nu},\tag{B}$$
which is eq. (3).
The refractive index $n$ is by definition the reciprocal phase velocity. Assuming no dispersion, we can identify it with the reciprocal group velocity
$$|{\bf v}|~=~\frac{c}{n}.\tag{C}$$
Equivalently
$$\frac{1}{n^2}~\stackrel{(C)}{=}~\frac{g_{ab}\dot{x}^a\dot{x}^b}{-g_{00}(\dot{x}^0)^2},\qquad
\dot{x}^{\mu} ~=~\frac{dx^{\mu}}{d\lambda}.\tag{D}$$
Eq. (1.10) and the speed of light condition (C) suggests that
in adapted coordinates
$$ \dot{x}^a~=~ep^a, \qquad \dot{x}^0~=~ en^2p^0, \tag{E}$$
where $e$ is an einbein/Lagrange multiplier. In covariant form eq. (E) reads
$$ \frac{\dot{x}^{\mu}}{e}~=~p^{\mu} -(n^2-1)V^{\mu} (p\cdot V)~=~G^{\mu\nu} p_{\nu}, \tag{F}$$
where we have introduced an effective metric tensor
$$ G_{\mu\nu}~=~g_{\mu\nu} + (1-n^{-2}) V_{\mu}V_{\nu}\qquad\Leftrightarrow\qquad G^{\mu\nu}~=~g^{\mu\nu} - (n^2-1) V^{\mu}V^{\nu}. \tag{G} $$
We next consider the Hamiltonian Lagrangian
$$L_H~=~p\cdot \dot{x}- H,\tag{H}$$
where the Hamiltonian is of the form Lagrange multiplier times constraint
$$ H~:=~\frac{e}{2} p_{\mu}G^{\mu\nu}p_{\nu}
~=~\frac{e}{2}\left( p^2 - (n^2-1) (p\cdot V)^2 \right).\tag{I}$$
In the gauge $e=1$ the Hamiltonian $H$ becomes eq. (17) in Ref. 3 and eq. (4) in Ref. 1. Note that eq. (4) contains a sign mistake in the second term.
The EL equations of $L_H$ wrt. $p_{\mu}$ and $e$ reproduce eq. (F) and eq. (1.10), respectively, as they should.
The Hamiltonian action principle
$$ S_H ~=~\int d\lambda~L_H \tag{J}$$
agrees with the variational principle (16) in Ref. 3.
It is straightforward to integrate out $p_{\mu}$ and/or $e$ to arrive at corresponding Lagrangian formulations, cf. e.g. this Phys.SE post. The light trajectories are null-like geodesics wrt. the $G_{\mu\nu}$ metric.
References:
A. Rogers, Frequency-dependent effects of gravitational lensing within plasma, arXiv:1505.06790.
S.M. Barnett & R. Loudon, The enigma of optical momentum in a medium, Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences 368 (2010).
O.Yu. Tsupko & G.S. Bisnovatyi-Kogan, Gravitational lensing in plasma: Relativistic images at homogeneous plasma, arXiv:1305.7032.
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$^1$ Ref. 2 claims in section 5 that the resolution of the controversy is that the Abraham 3-momentum is the kinetic momentum of the light in the medium, while the Minkowski 3-momentum is the canonical momentum.
$^2$ Let $\mu,\nu\in\{0,1,2,3\}$ be spacetime indices while $a,b\in\{1,2,3\}$ are spatial indices.
