Having mulled this over for a while longer in my own head, and building on the reasoning in Willk's answer and Nicol Wollaston's answer, I think I can now definitively say "Yes! A joint being spun by a 3D arrangement of fixed muscle attachments can indeed drive a continuously-rotating shaft extending in a 4th dimension."
We start by considering rotation in a plane. There is a single stationary point at the center of rotation. Adding a third dimension, we can extend this point into a stationary line, or axis. By itself, that line cannot transmit rotational motion; in order to generate torque, you must have an attachment point with some finite offset in a direction parallel to the plane of rotation--or, in other words, a driveshaft centered on the axis must have non-zero thickness.
Now, here's the key part: consider a different third dimension. If we are working in a 4D space, either of two dimensions perpendicular to the plane will work for extending a shaft. The two dimensions are indistinguishable, in that either of them can form a 3D subspace of the 4D world in combination with the plane of rotation, and either one can have lines extended from it parallel to that plane defining the radius of a driveshaft. In fact, in general, we could have an entire drivesheet, or an arbitrary number of driveshafts all intersecting and rigidly connected to the plane of rotation at the same place, but rotated at arbitrary angles between the orthogonal cardinal directions of whatever coordinate system we choose.
Now, we just have to determine whether or not such a driveshaft must necessarily intersect the belts/muscles/whatever that attach to the joint at some point during a full double-rotation. And in fact, we can show that it need not do so. First, note that the motion of the belts can be confined entirely to a 3D hyperplane--we know this because the belt mechanism works in 3D! All of the components will occupy some 4D depth, but only a finite 4D depth. The driveshaft must also have finite radius in all 3 dimensions perpendicular to its axis--this means that it will have some finite extent in the dimension used to defined the 3D hyperplane within which the belt arrangement operates. However, we can also arrange that, as each belt passes over alternating faces of the central cube (or hypersquare prism, hypercylinder, or whatever), it can be made to pass arbitrarily close to the surface of that joint. Now, we imagine the driveshaft attached to the center of the 3D hypersurface of the joint; supposing we have belts of thickness $t$, we can arrange that no part of a belt extends further from the surface of the joint than distance $t$ by building the entire mechanism compactly, so that the belts are always sliding in contact with the surface of the joint. If the driveshaft has a radius of $r$, then as long as the joint itself has a minimum radius of $t+r$, there is no way that the shaft can ever come into contact contact with the belts, let alone intersect their paths.
I strongly suspect that the reality of the situation is actually much looser than that, but that much I feel I can prove. In fact, it should be possible to attach a nearly-complete 2D drivesheet to the joint, with mere holes cut out to permit the passage of muscle belts. When viewing only the 3D hyperplane containing the muscle belts, the intersection of the drivesheet with that subspace would appear as a set of disconnected axle sections floating unsupported, with the belts passing through the gaps in between.