An estimate would be from trajectory simulation, but that doesn't seem satisfying...
Why doesn't that seem very satisfying? That is exactly what was done, many times over.
From the perspective of the New Horizons operations team, they didn't need to know where Pluto or Arrokoth were 100 years ago, or where they will be 100 years from now. They needed to know
- When New Horizons would make its closest approach to the target,
- At what distance that would occur,
- How much this deviated from the desired result, and
- What minimal trajectory correction maneuvers would correct the deviation.
The margins were very tight, no more than 150 seconds error (one sigma) for the time of closest approach and no more than 100 kilometers for the closest approach distance. The uncertainties in even the best available solar system and Pluto system ephemerides did not come close to meeting these needs. The situation with Arrokoth was even worse.
That New Horizons did not have a radar altimeter was irrelevant; a radar altimeter wouldn't have helped. By the time New Horizons got close enough where a radar altimeter would have been useful for measuring distance, it would have been far too late to do anything. What New Horizons did have that was useful was its cameras.
New Horizons was taken out of hibernation mode well in advance of the encounters with Pluto and Arrokoth (over six months for Pluto and over four months for Arrokoth). At these points the vehicle began taking series of images of the target. The relayed imagery in combination with estimations of the position and velocity state of New Horizons enabled the navigation team on Earth to update its estimate of the target's orbit, with a sharp focus on short term accuracy.
Along the way, the navigation team simulated the vehicle's and the target's trajectories, many times over.
The navigation team used the same technique to provide post-flyby estimates the time of and distance to the target at closest approach. The only differences between the pre- and post- flyby results are that
- The post-flyby navigated state included imagery from both before and after the flyby (including some from very close to the closest approach), and
- The post-flyby estimate of the time of and the distance at closest approach is an interpolated rather than an extrapolated result.
The combination of the above inevitably made the post-flyby estimates better of the closest approach time and distance more accurate than the pre-flyby estimates, but the basic technique remained the same.
Primary source: Williams, B., et al. "Navigation Strategy and Results for New Horizons’ Approach and Flyby of the Pluto System." AAS/AIAA Astrodynamics Specialist Conference Vale, CO. 2015.
The conference at which this paper was presented is informally known as the GN&Ski conference. It's held once a year, in winter, and at a ski resort. Morning conference sessions start ridiculously early, and afternoon conference sessions run well into the evening. But there's a nice long gap between the morning and afternoon sessions that gives attendees a bit of time to hit the slopes.