= Solution
Put $T=t-t'$. <momentum> <eigenstates> evolve with <energy> $p^2/(2m)$, so the retarded <momentum> kernel is $\theta(T)\delta(p-p')e^{-ip^2T/(2m\hbar)}$. Its <Fourier transform> reduces to
$$
K_0=\frac{\theta(T)}{2\pi\hbar}\int dp\,
\exp\left[\frac i\hbar\left(p(x-x')-\frac{p^2T}{2m}\right)\right].
$$
Completing the square and defining the oscillatory <Gaussian integral> by a convergent limiting prescription gives
$$
\boxed{K_0=\theta(T)\sqrt{\frac{m}{2\pi i\hbar T}}
\exp\left[\frac{im(x-x')^2}{2\hbar T}\right]}.
$$
For $T>0$ the <square root> has <quantum phase> $e^{-i\pi/4}$. As $T\downarrow0$, this kernel tends to $\delta(x-x')$ in the distributional sense. Differentiating the <Heaviside step function> consequently supplies the required $i\hbar\delta(T)\delta(x-x')$ source; an extra $-i/\hbar$ factor would correspond to a different Green-function convention.
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