Is there a single term that completes this sentence: "at its _____, Pluto is about 27AU from Earth." peri- something...?
Usually the term 'closest approach' is used. It applies to any two bodies in motion. Perigee is a term occasionally used, but the dictionaries say it only applies to things orbiting Earth. Use your discretion.
In the case of the planets that orbit farther from the sun than us their closest approaches occurs when they are on the opposite side of Earth that the sun is, which is known as an opposition. For Venus and Mercury, it happens when they align with the sun from our viewpoint, and that is known as an inferior conjunction. From Wikipedia - Opposition (planets):
Pluto has an inclination of 17o to the ecliptic, and an orbital eccentricity of 0.25. Either one of those things is enough to mean that when Pluto is closest to the sun, that brings it much closer to us, and the change in distance between when we are in front of or behind the sun in relation to it each year is a much smaller factor by comparison. For instance, right now it is about 33 AU from the sun, while on average it is 39 AU. When it is in opposition, that puts us about 32 AU away, and 34 AU when it is in conjunction. However, the orbit of Pluto is 248 years, so for all practical purposes the point each year when it is in opposition is what matters, unless you are willing to wait for as much as that time for it to get closer. The same holds true for all the possible dwarf planets in the Kuiper Belt.
Not quite what you want, but very closely related to what you want are the terms "opposition" (for the outer planets) and "inferior conjunction" (for Venus and Mercury).
Perigee would be the closest, but that may be abusing the term. Perigee means "closest to Earth", but generally it's used for objects that are in Earth's orbit.
MOID. Minimum Orbit Intersection Distance.
Usually used for Near Earth Objects when assessing risk of impact.
But Wikipedia writes "It is defined as the distance between the closest points of the osculating orbits of two bodies. " So perhaps MOID could have a more general use than just the minimum distance between an NEA and earth.
In general the minimum distance to an outer body would occur at opposition and when the outer body is at perihelion. Minimum distance to an inner body would occur at inferior conjunction when inner body is at aphelion. But even these general rules could have exceptions if the orbits are inclined to one another and/or resonant.
For example a Hilda asteroid might have a 5.19 A.U. aphelion and Jupiter's orbit is 5.2 A.U.. Does this mean that a Hilda could get within .01 of Jupiter? No. Because of a Hilda's resonance with jupiter, aphelions will only occur near Jupiter's L4, L5 and L3 regions. Finding a Hilda Jupiter MOID is an interesting geometry problem. I suspect it doesn't occur at opposition or inferior conjunction. If I have time and energy to play with this puzzle, I'll may add to my answer