  Using the phat-ELVIS suite of Milky Way-size halo simulations, we show that
subhalo orbital pericenters, $r_{\rm peri}$, correlate with their dark matter
halo structural properties. Specifically, at fixed maximum circular velocity,
$V_{\rm max}$, subhalos with smaller $r_{\rm peri}$ are more concentrated (have
smaller $r_{\rm max}$ values) and have lost more mass, with larger peak
circular velocities, $V_{\rm peak}$, prior to infall. These trends provide
information that can tighten constraints on the inferred $V_{\rm max}$ and
$V_{\rm peak}$ values for known Milky Way satellites. We illustrate this using
published pericenter estimates enabled by Gaia for the nine classical Milky Way
dwarf spheroidal satellites. The two densest dSph satellites (Draco and Ursa
Minor) have relatively small pericenters, and this pushes their inferred
$r_{\rm max}$ and $V_{\rm max}$ values lower than they would have been without
pericenter information. For Draco, we infer $V_{\rm max} = 23.5 \, \pm 3.3$ km
s$^{-1}$ (compared to $27.3 \, \pm 7.1$ km s$^{-1}$ without pericenter
information). Such a shift exacerbates the traditional Too Big to Fail problem.
Draco's peak circular velocity range prior to infall narrows from $V_{\rm peak}
= 21 - 49$ km s$^{-1}$ without pericenter information to $V_{\rm peak} = 25-37$
km s$^{-1}$ with the constraint. Over the full population of classical dwarf
spheroidals, we find no correlation between $V_{\rm peak}$ and stellar mass
today, indicative of a high level of stochasticity in galaxy formation at
stellar masses below $\sim 10^7$ M$_\odot$. As proper motion measurements for
dwarf satellites become more precise, they should enable useful priors on the
expected structure and evolution of their host dark matter subhalos.
