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 $$\sum_{i=1}^3 \frac{x_i^2}{a_i^2 + \lambda} = 1$$