According to this article on the spacetime algebra, we know the Dirac spinor can be thought of as an even element of the Clifford algebra over spacetime, which in turn can be thought of as a general transformation on bivectors, since it consists of a dilation (scalar), Lorentz transformation ("rotor"), and duality rotation (complex phase).
Thus, the presence of the $U(1)$ gauge field apparently speaks to the duality rotation specifically.
Also, we know that $SU(2)$ is isomorphic to the 3D rotors, hence spatial rotations. The double cover is nothing to worry about — it's just because a rotor gets both pre- and post-multiplied (just like the wavefunction itself in many cases, so that matches) in order to effect a rotation, so the angle of the rotor gets doubled into the rotation angle.
So what about $SU(3)$? It is 8-dimensional, just like the even sub-algebra itself, which might make me think it could represent an entire bivector transformation, but that doesn't make sense because it is normalized whereas the dilation part needs to range to infinity. That is, the even sub-algebra really has 7 rotation-type DOF and one magnitude.
I know that phrasing these gauge operations in GA language might end up going nowhere, but I found the interpretations of the first two symmetries to be interesting, so I was wondering if something in that spirit could be found for $SU(3)$ as well.