= Solution
There is a sign inconsistency in the original PDF: varying its action gives a plus sign for the spatially projected magnetic term, whereas its displayed target equation has a minus sign. The following <Lagrange multiplier for antisymmetry> calculation derives the result of the action as printed.
Temporarily vary all sixteen entries $M^{\mu\nu}$ independently and add
$$
\mathcal L_{\lambda}=\frac12\lambda_{\mu\nu}(M^{\mu\nu}+M^{\nu\mu}),\qquad \lambda_{\mu\nu}=\lambda_{\nu\mu}.
$$
Variation of the symmetric <Lagrange multiplier> imposes $M^{\mu\nu}+M^{\nu\mu}=0$. Define
$$
w_\mu=u_\rho M^\rho{}_{\mu},\qquad C_{\mu\nu}=h_{\mu\rho}h_{\nu\sigma}M^{\rho\sigma}.
$$
On this constraint surface, $C_{\nu\mu}=-C_{\mu\nu}$. Differentiating the original contractions before imposing the constraint gives
$$
\left.\frac{\partial}{\partial M^{\mu\nu}}(u_\alpha u_\beta M^{\lambda\alpha}M^\beta{}_{\lambda})\right|_{M=-M^{\mathsf T}}
=u_\nu w_\mu-u_\mu w_\nu,
$$
and
$$
\left.\frac{\partial}{\partial M^{\mu\nu}}(h_{\alpha\rho}h_{\beta\sigma}M^{\alpha\beta}M^{\sigma\rho})\right|_{M=-M^{\mathsf T}}
=-2C_{\mu\nu}.
$$
In particular, the reversed order $\sigma\rho$ in the last factor of the action matters. The unrestricted <Euler-Lagrange field equation>, evaluated on the antisymmetry constraint, is
$$
0=\frac12F_{\mu\nu}-\frac1{2\kappa}(u_\nu w_\mu-u_\mu w_\nu)+\frac1{2\chi}C_{\mu\nu}+\lambda_{\mu\nu}.
$$
Subtract its index-transposed equation to eliminate the symmetric multiplier:
$$
\boxed{0=F_{\mu\nu}-\frac1\kappa\left(u_\nu u_\rho M^\rho{}_{\mu}-u_\mu u_\rho M^\rho{}_{\nu}\right)+\frac1\chi h_{\mu\rho}h_{\nu\sigma}M^{\rho\sigma}.}
$$
A direct check is to restrict the <action> to a rest-frame magnetic component $M^{12}=m$, $M^{21}=-m$: its quadratic contribution is $+m^2/(2\chi)$, whose derivative has a positive sign. To obtain the PDF's target equation instead, one must reverse the sign of that quadratic term in the action, or replace its last $M^{\sigma\rho}$ by $M^{\rho\sigma}$. These are alternative repairs; changing conventions cannot make the printed action and printed equation agree.
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