  We investigated the Gilbert damping in La$_{0.7}$Sr$_{0.3}$MnO$_3$ (LSMO) and
La$_{0.7}$Sr$_{0.3}$MnO$_3$/Pt (LSMO/Pt) heterostructures using
first-principles calculations and Wannier interpolation techniques. Our work is
motivated by recent experimental observations showing smaller Gilbert damping
in LSMO/Pt films compared to their reference single-layer LSMO films, despite
expectations of enhanced spin-pumping effects in the former. We analyze the
electronic structures and transport behaviors, finding that LSMO thin films
have a high spin Hall angle ($|{\theta_{\mathrm{SH}}}|$). However, in LSMO/Pt,
the presence of platinum significantly increases longitudinal conductivity,
reducing $|\theta_{\mathrm{SH}}|$. Despite the lower $|\theta_{\mathrm{SH}}|$,
LSMO/Pt shows a notable anti-damping contribution to Gilbert damping due to a
larger spin diffusion length. In contrast, pure LSMO films with large
$|\theta_{\mathrm{SH}}|$ exhibit higher damping due to efficient spin-to-charge
conversion via a self-induced inverse spin Hall effect (ISHE), as reported in a
recent experiment. Finally, this work demonstrates that by fine-tuning the
ratio of spin Hall conductivity to longitudinal charge conductivity, it is
possible to engineer heterostructures with desired spin-to-charge or
charge-to-spin conversion efficiencies even with weaker spin-orbit couplings.
