= Solution
Use the <Minkowski metric> convention $\eta=\operatorname{diag}(1,-1,-1,-1)$. On the <Adjoint representation>, the <adjoint covariant derivative> is $D_\mu X=\partial_\mu X+[A_\mu,X]$. Set $a_\mu=\delta A_\mu$ and $\chi=\delta\Phi$. Then
$$
\delta F_{\mu\nu}=D_\mu a_\nu-D_\nu a_\mu,\qquad
\delta(D_\mu\Phi)=D_\mu\chi+[a_\mu,\Phi].
$$
The <trace> obeys $\operatorname{Tr}(X[Y,Z])=\operatorname{Tr}([Z,X]Y)$, and covariant <integration by parts> has the ordinary boundary term because the <trace> of a <commutator> is zero. Thus compactly supported variations give
$$
\begin{aligned}
\delta S=\int d^4x\,\operatorname{Tr}\bigl(&F^{\mu\nu}D_\mu a_\nu
-D^\mu\Phi\,D_\mu\chi-D^\mu\Phi[a_\mu,\Phi]+m^2\Phi\chi\bigr)\\
=\int d^4x\,\operatorname{Tr}\bigl(&[-D_\mu F^{\mu\nu}+[D^\nu\Phi,\Phi]]a_\nu
+[D_\mu D^\mu\Phi+m^2\Phi]\chi\bigr).
\end{aligned}
$$
The <trace> pairing on $\mathfrak{su}(2)$ is nondegenerate, so the <Euler-Lagrange field equations> are
$$
\boxed{D_\mu F^{\mu\nu}=[D^\nu\Phi,\Phi],\qquad D_\mu D^\mu\Phi+m^2\Phi=0.}
$$
In particular the sign of the scalar mass term follows from the plus sign in the supplied <trace> <Lagrangian density>; it should not be guessed from a different metric convention. This is the <adjoint Yang-Mills-Higgs trace variation>.
For the static <Bogomolny-Prasad-Sommerfield monopole>, take the usual purely magnetic extension $\partial_0A_i=\partial_0\Phi=0$ and <temporal gauge> $A_0=0$. At $m=0$, the equations reduce to
$$
D_jF_{ji}=[\Phi,D_i\Phi],\qquad D_iD_i\Phi=0,
$$
with the $\nu=0$ equation identically zero. We now verify both spatial equations from $B_i=D_i\Phi$ without assuming them. The <gauge-theory Bianchi identity> gives $D_iB_i=0$, hence immediately $D_iD_i\Phi=0$. Also $F_{ji}=\varepsilon_{jik}B_k$, so
$$
\begin{aligned}
D_jF_{ji}&=\varepsilon_{jik}D_jD_k\Phi
=\frac12\varepsilon_{jik}[D_j,D_k]\Phi\\
&=\frac12\varepsilon_{jik}[F_{jk},\Phi]
=-[B_i,\Phi]=[\Phi,D_i\Phi].
\end{aligned}
$$
Here $\varepsilon_{jik}\varepsilon_{jk\ell}=-2\delta_{i\ell}$. This proves that the <Bogomolny equations> imply all static zero-mass <Yang-Mills equations> and the scalar equation. The specification of $A_0$ matters: the spatial <Bogomolny equations> by themselves do not constrain an arbitrary extra electric potential, so the implication is for their standard purely magnetic static extension.
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