= Solution
At $\Omega=\kappa$, the inverse-transform denominator is $-(\omega-i\kappa)^2$. For $t>0$, its upper-half-plane double-pole <residue> is the <derivative> of the exponential:
$$
G(t)=-i\left.\frac{d}{d\omega}e^{i\omega t}\right|_{\omega=i\kappa}
=\boxed{t e^{-\kappa t}}.
$$
It is also the continuous limit of both answers above as $\beta\to0$ or $\eta\to0$. This <critically damped> response tends to zero with a linear factor multiplying the exponential.
In all three cases, for $t<0$ close the <contour> in the lower half-plane, where $e^{i\omega t}$ decays. There are no <poles> inside, so
$$
\boxed{G(t)=0\quad(t<0)}.
$$
At $t=0$ its continuous value is zero and $G'(0^+)=1$, producing the unit <derivative> jump required by $(D_t^2+2\kappa D_t+\Omega^2)G=\delta(t)$. Thus all three formulas are causal <Green's functions> and all decay to zero at late times.
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