The gravitational potential of ellipsoid

This is a very late answer. There is the book

Ellipsoidal Figures of Equilibrium

by the god himself in this field S. Chandrasekhar. Chapter 3 is devoted fully to understanding the gravitational potentials of ellipsoids. Theorems 3 and 9 are what you are looking for.

You also asked

Can they be derived in terms of ellipsoidal harmonics?

Chandrasekhar does not derive the equations in terms of ellipsoidal harmonics. In fact, he states that very early on in the introduction (section 16). Instead he employs spherical polar coordinates and proceeds by establishing a series of lemmas on the moments of the mass distribution. This amounts to considering integrals of the form $$I(u) = a_1 a_2 a_3 \int_u^{\infty} \frac{du}{\Delta}; \qquad A_i(u) = a_1 a_2 a_3 \int_u^{\infty} \frac{du}{\Delta (a_i^2 + u)}$$ where $\Delta^2=(a_1^2+u) (a_2^2+u) (a_3^2+u)$ and $a_i$ are the semi-major axes of the ellipsoid. Then come the two theorems you need

Theorem 3: At a point $x_i$ interior to the ellipsoid, the potential is $$\Phi = \pi G \rho \Big[I(0) - \sum_{i=1}^3 A_i(0) x_i^2 \Big]$$

Theorem 9: At a point $x_i$ exterior to the ellipsoid, the potential is $$\Phi = \pi G \rho \Big[I(\lambda) - \sum_{i=1}^3 A_i(\lambda) x_i^2 \Big]$$ where $\lambda$ is the (algebraically) largest root of $$\sum_{i=1}^3 \frac{x_i^2}{a_i^2 + \lambda} = 1$$