Well, I sort-of solved it! I noticed that the designers had also created a "stp" file of the part. I imported it into the figure, selected Tools, selected the part, selected Axis, selected part of the circumference of the "circle" that I wanted it to rotate around, clicked Add, selected another part of the circumference, clicked OK, and the axis appeared. Works perfectly!
Then I selected and deleted the original prt file and moved the stp file in the sBOM to the desired spot.
The "only" detail is that if I move the part, the axis remains where it was, rather than moving along with the part! Is there any way to "group" the axis to the part?