You can use an infinite sequence of image charges, alternating between the interiors of the two spheres. It's like the limiting problem of two infinite parallel planes, except that the positions and magnitudes of the successive charges are more complicated. However, the asymptotic behavior of the infinite series should be (relatively) straightforward to determine.

With two spheres, it's easy to write an explicit Green's function as an infinite series, and the behavior of that ought to tell you what you need to know about the solution.

If there were something like the $\sin(\pi z)/\pi z$ formula for $\prod_{k=2}^\infty \left(1-\frac{z^{n}}{k^{3}}\right)$ it would provide a value for $\zeta(3)$.