  We investigate the endofullerene system $^3$He@C$_{60}$ with a
four-dimensional Potential Energy Surface (PES) to include the three He
translational degrees of freedom and C$_{60}$ cage radius. We compare MP2,
SCS-MP2, SOS-MP2, RPA@PBE and C(HF)-RPA to calibrate and gain confidence in the
choice of electronic structure method. Due to the high cost of these
calculations, the PES is interpolated using Gaussian Process Regression (GPR),
owing to its effectiveness with sparse training data. The PES is split into a
two-dimensional radial surface, to which corrections are applied to achieve an
overall four-dimensional surface. The nuclear Hamiltonian is diagonalised to
generate the in-cage translational/vibrational eigenstates. The degeneracy of
the three-dimensional harmonic oscillator energies with principal quantum
number $n$ is lifted due to the anharmonicity in the radial potential. The
$(2l+1)$-fold degeneracy of the angular momentum states is also weakly lifted,
due to the angular dependence in the potential. We calculate the fundamental
frequency to range between 96cm$^{-1}$ and 110cm$^{-1}$ depending on the
electronic structure method used. Error bars of the eigenstate energies were
calculated from the GPR and are on the order of approximately $\pm$
1.5cm$^{-1}$. Wavefunctions are also compared by considering their overlap and
Hellinger distance to the one-dimensional empirical potential. As with the
energies, the two \textit{ab initio} methods MP2 and RPA@PBE show the best
agreement. While MP2 has better agreement than RPA@PBE, due to its higher
computational efficiency and comparable performance, we recommend RPA as an
alternative electronic structure method of choice to MP2 for these systems.
