When Neptune and Pluto are closest, about 100 million mi (160 million km) from each other, would an observer on Neptune (or rather on one of its moons, since Neptune is gaseous) be able to see Pluto, and maybe even Charon, with the naked eye? If not, could Pluto be seen in average binoculars? I think Pluto would appear a bit smaller than Mercury from Earth (but at much lower apparent brightness because of the distance from the Sun).
No, it cannot. Far from it.
The closest approach between both planets is roughly 16 AU due to the 3:2 orbit resonance. Pluto will even then be a tiny dot among many with a brightness around 14 mag.
You can try that with Stellarium yourself, placing the observer on Neptune and looking for Pluto. You just have to find the right time. One such time is approx. in the year 2877.
According to the NASA Pluto fact sheet, the brightest that Pluto gets is an apparent visual magnitude of 13.65 when it is 28.6 au from the Earth and (presumably) about 29.6 au from the Sun.
To work out how bright that would be from Neptune we could work out how close Neptune can be to Pluto when Pluto is at perihelion. This is complicated by the fact that the semi-major axes of the orbits are not aligned, that the inclination of Pluto to the ecliptic is quite high and that the orbital periods are in a 3:2 resonance. The closest they can actually get is about 16 au and this will not occur when Pluto is at perihelion. And, when Pluto is at perihelion it will be further than 16 au from Neptune and will not be fully illuminated (or even as illuminated as we can see it from Earth) as seen from Neptune.
Therefore an upper limit to the brightness would be to scale how bright it can appear from Earth to a distance of 16 au, which gives an apparent magnitude $>12.4$.
For the reasons stated above it must be fainter than this because it is not fully illuminated as seen from Neptune at 16 au and it is not at perihelion at that time, and so it cannot be anywhere near visible to the naked eye.
EDIT: A paper by Williams & Benson (1971) puts the closest approach of Pluto and Neptune at 16.8 au, but crucially, this occurs when Pluto is near aphelion. They also show that this is true for a timespan exceeding 1 million years.
That's great question! ... I'm just thinking that just for comparison, Pluto is roughly 70% the diameter of our Moon and Pluto would be far away from Neptune than the Moon is from Earth. Moon is also sort of "fixed" to Earth, but Pluto is in a different orbit than Neptune. I think that Pluto will not be visible with the naked eye given its size and distance from the observer.