= Solution
Raising each beam energy to 250 MeV makes $E=500$ MeV. The <kaon> denominator is now dominated by
$$
(E^2-m_K^2)^2=(500^2-498^2)^2=(1996\ \mathrm{MeV^2})^2,
$$
rather than by $E^2\Gamma_K^2$. Substituting into <2/c/a> gives
$$
\boxed{\mathcal R_{\rm inclusive}(500)=5.606\times10^{22},\qquad
\mathcal R_{\pi\pi}(500)=8.125\times10^{22}.}
$$
The enormous change is a tiny-width effect, not a comparably abrupt change in the broad rho contribution. The center-of-momentum energy has moved about $2/(7.4\times10^{-12})\simeq2.7\times10^{11}$ <kaon> widths from the peak. A width of $7.4\times10^{-12}$ MeV is only $7.4\times10^{-6}$ eV, requiring an energy-selection scale of order $10^{-14}$ relative to the several-hundred-MeV beam energy.
Real beams have vastly broader energy distributions. Averaging over such a distribution suppresses the useful narrow-resonance contribution: the finite peak value does not imply a large energy-integrated formation rate, whose resonant area shrinks with the width at fixed <branching fractions>. The weak inverse-decay entrance coupling and the broad strong-interaction rho background therefore make this a poor way to study <kaon> resonant formation. The supplied broad-resonance formula is used algebraically as requested; a practical calculation would convolve an appropriate narrow line shape with the actual beam distribution.
\Image[/past-exam-of-the-sub-atomic-physics-course-of-the-university-of-oxford/2016-broad-rho-and-narrow-kaon-resonances.png]
{title=A broad rho background and an extremely narrow kaon resonance}
{description=Original evaluation of the stipulated line shapes. The kaon inset uses its own width-scale horizontal axis; drawing it visibly broad on a MeV axis would misrepresent the given width.}
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