  Analysis of inclusions in primitive meteorites reveals that several
short-lived radionuclides (SLRs) with half-lives $0.1-100$ Myr existed in the
early Solar System (ESS). We investigate the ESS origin of $^{107}$Pd,
$^{135}$Cs, and $^{182}$Hf, which are produced by $slow$ neutron captures (the
$s$-process) in asymptotic giant branch (AGB) stars. We modelled the galactic
abundances of these SLRs using the OMEGA+ galactic chemical evolution (GCE)
code and two sets of mass- and metallicity-dependent AGB nucleosynthesis yields
(Monash and FRUITY). Depending on the ratio of the mean life $\tau$ of the SLR
to the average length of time between the formation of AGB progenitor $\gamma$,
we calculate timescales relevant for the birth of the Sun. If
$\tau/\gamma\gtrsim2$, we predict self-consistent isolation times between 9 and
26 Myr by decaying the GCE predicted $^{107}$Pd/$^{108}$Pd,
$^{135}$Cs/$^{133}$Cs, and $^{182}$Hf/$^{180}$Hf ratios to their respective ESS
ratios. The predicted $^{107}$Pd/$^{182}$Hf ratio indicates that our GCE models
are missing $9-73\%$ of $^{107}$Pd and $^{108}$Pd in the ESS. This missing
component may have come from AGB stars of higher metallicity than those that
contributed to the ESS in our GCE code. If $\tau/\gamma\lesssim0.3$, we
calculate instead the time ($T_{\rm LE}$) from the last nucleosynthesis event
that added the SLRs into the presolar matter to the formation of the oldest
solids in the ESS. For the 2 M$_\odot$, $Z=0.01$ Monash model we find a
self-consistent solution of $T_{\rm LE}=25.5$ Myr.
