= Solution
Subtract the <reciprocal lattice vector> $(0,0,8)$, leaving a reduced <wave vector> $(0.2,0,0)$ along $a$. A linear-acoustic estimate gives
$$
q=0.2\frac{2\pi}{0.353\ \mathrm{nm}}=3.560\times10^9\ \mathrm{m^{-1}},\qquad
v_g\simeq\frac{E}{\hbar q}=2.99\times10^3\ \mathrm{m\,s^{-1}}.
$$
Interpret the quoted intrinsic width as the full width at half maximum in energy, $\Gamma=0.1$ meV. The corresponding energy-relaxation <mean lifetime> is $\tau=\hbar/\Gamma=6.58\times10^{-12}$ s, giving <mean free path> $v_g\tau\simeq19.7$ nm. If the given width instead denotes a half width, the inferred <mean lifetime> differs by a factor of two; the convention must be specified.
Approximating the thermal average by this velocity and <mean lifetime>, and using the isotropic angular factor $\langle v_a^2\rangle=v_g^2/3$, kinetic theory gives
$$
\boxed{\kappa_{{\rm ph},a}\simeq\tfrac13 C_Vv_g^2\tau
=0.72\ \mathrm{W\,m^{-1}\,K^{-1}}.}
$$
Using a single purely $a$-directed effective channel without angular averaging gives $2.17$ W m$^{-1}$ K$^{-1}$, still far below the observed total $39$ W m$^{-1}$ K$^{-1}$. The isotropic estimate is roughly $54$ times smaller than the measured total.
The calculation is not a complete thermal-transport average. Electrons in this metallic compound can carry substantial heat, the supplied total <heat capacity> need not be entirely phononic, and one $7$ meV <phonon> need not represent all thermally active branches and <mean lifetimes>. At $20$ K, $k_BT\simeq1.72$ meV, so that measured mode is relatively high in the thermal distribution; lower-energy modes can have much longer <mean lifetimes>. A measured <phase velocity> is also only an estimate of the <group velocity> if the branch is nonlinear.
<electron-phonon interaction> is a plausible damping mechanism in a phonon-mediated <superconductor>, but the measurement is \b[above] its $15$ K transition and probes the normal state. It cannot be attributed directly to decay into an already established superconducting condensate. Below the transition an energy gap can modify electron-hole damping. The weak-coupling scale $2\Delta\simeq3.52k_BT_c\simeq4.55$ meV is below the $7$ meV <phonon> energy, so even superconductivity need not prohibit all pair-breaking damping of this mode. Temperature-dependent spectra would distinguish that mechanism from defects or anharmonic <phonon> decay.
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