Method of images / charge density uniqueness

But how can a collection of point charges recreate a continuous charge distribution? The method of images emphasizes the uniqueness of the solutions, and I think that because $V$ is uniquely determined then $rho$ must be as well.

The reasoning is a bit more complex than that. Uniqueness tells us that if you specify the potential on a surface, and demand that there is no charge outside the surface, then the potential outside the surface is uniquely determined. And indeed, both the actual charge distribution and the image charges produce the same potential outside, even though they have different charge distributions and potentials inside.

Now, using the image charge configuration, you can easily determine the potential outside. So to get the actual charge distribution, you additionally impose the condition that the potential is uniform inside. With this additional condition, you now have the potential everywhere, which determines the charge distribution everywhere. In other words, all the image charge trick is doing is giving you an easy way to compute the potential outside, which is the part that's actually unique.

Incidentally, the fact that a charge distribution inside a surface isn't determined uniquely by boundary conditions on a surface should be very familiar. The shell theorem says that the potentials outside a uniformly charged spherical shell, and a point charge at the center, are identical.