We develop a uniform semiclassical trace formula for the density of states of a three-dimensional isotropic harmonic oscillator (HO), perturbed by a term 1/4 epsilon r(4). This term breaks the U(3) symmetry of the HO, resulting in a spherical system with SO(3) symmetry. We first treat the anharmonic term for small e in semiclassical perturbation theory by integration of the action of the perturbed periodic HO orbit families over the manifold CP2 which is covered by the parameters describing their four-fold degeneracy. Then, we obtain an analytical uniform trace formula for arbitrary E which in the limit of strong perturbations (or high energy) asymptotically goes over into the correct trace formula of the full anharmonic system with SO(3) symmetry, and in the limit E (or energy) -> 0 restores the HO trace formula with U(3) symmetry. We demonstrate that the gross-shell structure of this anharmonically perturbed system is dominated by the two-fold degenerate diameter and circular orbits, and not by the orbits with the largest classical degeneracy, which are the three-fold degenerate tori with rational ratios (omega(r) : omega(phi) = N : M of radial and angular frequencies. The same holds also for the limit of a purely quartic spherical potential V(r) proportional to r(4).