= Solution
Under the stipulated peak-energy approximation, set $E=m_V$ separately for $V=W,Z$ in the supplied <Breit-Wigner resonance cross section>. For a spin-one resonance,
$$
\sigma_V(m_V)\propto\frac{3B_{\rm in}^VB_{\rm out}^V}{m_V^2}
=\frac{3\Gamma_{\rm in}^V\Gamma_{\rm out}^V}{m_V^2\Gamma_V^2}.
$$
Define the measured ratio as $R=\sigma(W\to\ell\nu)/\sigma(Z\to\ell\ell)$. With the known production prefactor ratio neglected as instructed, the spin factors cancel and
$$
R=\frac{m_Z^2}{m_W^2}\frac{\Gamma_{\rm in}^W\Gamma_{\rm out}^W}
{\Gamma_{\rm in}^Z\Gamma_{\rm out}^Z}\frac{\Gamma_Z^2}{\Gamma_W^2}.
$$
Solving for the positive physical width gives
$$
\boxed{\Gamma_W=\Gamma_Z\frac{m_Z}{m_W}
\sqrt{\frac{\Gamma_{\rm in}^W\Gamma_{\rm out}^W}
{R\,\Gamma_{\rm in}^Z\Gamma_{\rm out}^Z}}.}
$$
If the production prefactors are retained as $C_W,C_Z$, multiply the quantity under the square root by $C_W/C_Z$. The <partial decay widths> in this extraction are calculated inputs, while the <total decay width> being inferred need not be fixed to its <Standard Model> value.
The <peak and integrated resonance width dependence> explains why the square root is essential to the specified \b[peak] approximation. An energy-integrated narrow-width cross section contains one inverse power of the <total decay width>, because integrating over the resonance supplies a factor proportional to its width. That different observable leads to a different extraction formula. Real hadron-collider analyses also include <parton distribution functions> and the energy convolution; the present result follows the explicit approximation in the source.
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