= Solution
Put $C=[A,B]$, which commutes with both operators by hypothesis. Differentiating $e^{\lambda A}Be^{-\lambda A}$ gives $C$, so this conjugate is $B+\lambda C$. Similarly the product $P=e^{\lambda A}e^{\lambda B}$ conjugates $A+B$ to itself: the two central commutator shifts cancel. Hence
$$
F'F^{-1}=A+(B+\lambda C)-(A+B)=\lambda C.
$$
With $F(0)=I$, the operator differential equation gives $F(\lambda)=e^{\lambda^2C/2}$. At $\lambda=1$ this proves the central-commutator <Baker--Campbell--Hausdorff formula>
$$
\boxed{e^Ae^B=e^{[A,B]/2}e^{A+B}}.
$$
These formal manipulations apply on a suitable common domain, or to the oscillator exponentials below where the corresponding unitary identities are well defined.
The oscillator has $[a,a^\dagger]=1$ and $H=\hbar\omega(a^\dagger a+1/2)$. Its normalized states and energies are
$$
\boxed{|n\rangle=\frac{(a^\dagger)^n}{\sqrt{n!}}|0\rangle,\qquad E_n=\hbar\omega(n+1/2)}.
$$
Since $\hat x=\sqrt{\hbar/(2m\omega)}(a+a^\dagger)$, put $\gamma=\mu\sqrt{\hbar/(2m\omega)}$. Taking $A=\pm i\gamma a^\dagger$, $B=\pm i\gamma a$ gives $[A,B]=\gamma^2$. The formula implies $e^{\pm i\mu\hat x}=e^{-\gamma^2/2}e^{\pm i\gamma a^\dagger}e^{\pm i\gamma a}$. Averaging the two signs proves the requested <normal ordering> of the cosine.
Because $a|0\rangle=0$, expansion of the remaining creation exponential gives
$$
\langle n|V|0\rangle=e^{-\gamma^2/2}\frac{(i\gamma)^n+(-i\gamma)^n}{2\sqrt{n!}}.
$$
Odd <matrix> elements vanish, the ground expectation is $e^{-\gamma^2/2}$, and the magnitude squared for $n=2p$ is $e^{-\gamma^2}\gamma^{4p}/(2p)!$. The nondegenerate <second-order nondegenerate perturbation theory> formula is $\Delta E_0=\epsilon V_{00}+\epsilon^2\sum_{n>0}|V_{n0}|^2/(E_0-E_n)+O(\epsilon^3)$. Therefore
$$
\boxed{\Delta E_0=\epsilon e^{-\mu^2\hbar/(4m\omega)}-
\frac{\epsilon^2e^{-\mu^2\hbar/(2m\omega)}}{\hbar\omega}
\sum_{p=1}^\infty\frac{(\mu^2\hbar/(2m\omega))^{2p}}{(2p)!(2p)}+O(\epsilon^3)}.
$$
The negative second-order sign comes from the higher unperturbed energies in every denominator.
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