= Solution
At the origin, the normalized <bound state in a delta potential> contributes $|\varphi_0(0)|^2=\kappa$, and the odd <scattering eigenfunctions> contribute zero. The even continuum functions from the preceding part have
$$
|\varphi_{k,e}(0)|^2=\frac1\pi\cos^2\delta_e=\frac1\pi\frac{k^2}{k^2+\kappa^2}.
$$
Their <energy> is $E_k=\hbar^2k^2/(2m)$. Inserting these weights into the <spectral resolution of a quantum propagator>, and using evenness to extend the integral to the whole real axis, yields for $T>0$
$$
\boxed{K(0,T|0,0)=\kappa e^{-iE_0T/\hbar}
+\int_{-\infty}^\infty\frac{dk}{2\pi}\frac{k^2}{\kappa^2+k^2}
e^{-i\hbar k^2T/(2m)}.}
$$
This is the <quantum propagator of an attractive delta potential>. The printed continuum exponent has an extra factor of $\hbar$: its time phase must be $-iE_kT/\hbar=-i\hbar k^2T/(2m)$. This correction also makes the exponent <dimensionless>. Oscillatory integrals are understood with an infinitesimal positive convergence factor before taking its limit.
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