= Solution
For a tail $\rho_\sigma\sim A_\sigma e^{-ar}$,
$$
|\nabla\rho_\sigma|\sim a\rho_\sigma,\qquad
x_\sigma\sim a\rho_\sigma^{-1/3},\qquad
\sinh^{-1}x_\sigma\sim ar/3.
$$
At large $r$, the B88 correction integrand becomes
$$
-\rho_\sigma^{4/3}
\frac{\beta x_\sigma^2}{6\beta x_\sigma\sinh^{-1}x_\sigma}
\sim-\frac{a\rho_\sigma}{6(ar/3)}
=-\frac{\rho_\sigma}{2r}.
$$
Summing the spin densities and dividing by total density gives
$$
\varepsilon_X^{\rm B88}\sim-\frac1{2r},\qquad
\boxed{c=\tfrac12}
$$
in <atomic units>. The exponentially decaying LDA term is negligible in this limit. The parameter $\beta$ cancels from the leading asymptotic term.
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