I haven't performed this operation yet - but the first workaround I'm going to try is as follows:
1) Use Generate Points Along Line to generate densely-spaced points along each 3D polyline, retaining the attributes of the corresponding line.
2) Use Intersect 3D to isolate the 3D points within my 3D multipatch sphere
3) Use the Points to Line tool to recreate 3D polylines from the 3D points, separating polylines based on an attribute.
Hope to test drive this in the next couple days. Would be happy to hear any improvements on this workflow if you have them!