  We report the discovery of HIP-97166b (TOI-1255b), a transiting sub-Neptune
on a 10.3-day orbit around a K0 dwarf 68 pc from Earth. This planet was
identified in a systematic search of TESS Objects of Interest for planets with
eccentric orbits, based on a mismatch between the observed transit duration and
the expected duration for a circular orbit. We confirmed the planetary nature
of HIP-97166b with ground-based radial velocity measurements and measured a
mass of $M_{b} =$ 20 $\pm$ 2 $M_\bigoplus$ along with a radius of $R_{b} =$ 2.7
$\pm$ 0.1 $R_\bigoplus$ from photometry. We detected an additional
non-transiting planetary companion with $M_{c}$ sin$i =$ 10 $\pm$ 2
$M_\bigoplus$ on a 16.8-day orbit. While the short transit duration of the
inner planet initially suggested a high eccentricity, a joint RV-photometry
analysis revealed a high impact parameter $b = 0.84 \pm 0.03$ and a moderate
eccentricity. Modeling the dynamics with the condition that the system remain
stable over $>$10$^5$ orbits yielded eccentricity constraints $e_b = 0.16 \pm
0.03$ and $e_c < 0.25$. The eccentricity we find for planet b is above average
for the small population of sub-Neptunes with well-measured eccentricities. We
explored the plausible formation pathways of this system, proposing an early
instability and merger event to explain the high density of the inner planet at
$5.3 \pm 0.9$ g/cc as well as its moderate eccentricity and proximity to a 5:3
mean-motion resonance.
