Can a probe orbit Pluto given Charon's gravity?

Avoiding hard math, which I'm not very good at, the simple answer is yes, provided there's enough distance. Pluto/Charon have 4 moons orbiting them a bit further away, all in relatively stable orbits. Source

Here's distances to scale. - same source.

Because the ratio of gravitational field strength to size is exponential (Power of 1.5), the apparent closeness of Pluto and Charon to the 4 small moons looks unusual, but it's perfectly acceptable for objects of that size.

Lagrange point stability requires a mass ratio of about 26 to 1. (.0385 to 1 per source). The Earth has satellites in unstable Lagrange points, so it's certainly doable, it's just not technically "stable".

The hardest part about getting a satellite to orbit Pluto is that Pluto's sufficiently small that any ship that approaches it would need to slow down significantly on it's own to get captured into a Pluto Orbit. That's why New Horizon was a flyby, not an orbit.