Hi Booleano,

You have several simple options to achieve your goal.

One could be to rebuild the path curve. Use the reconstructcurve script or the Rebuild command, specifying a large number of points.
Next your sweep will be possible without problem.
Why this method is a solution? Because your path is made with several small curves assembled. Rebuild and ReconstructCurve will recreate a single and continuous curve which is more simple to use by MoI.

A second option would be to extrude your shape.
Then, draw a line from the center of both ends of your extrusion.
Finally, use Transform/Deform/Flow, selecting your extrusion as the object, your line (the center line of the extrusion) as the source curve, and your desired path as the target curve.
Don't forget to check the "Stretch" box.

A small warning, those two possible methods will give you different results. Here the flowfunction will give you and object very large (with huge number of construct points).
Don't know why. No doubt, Michael could explain it.

Have fun with MoI modeling...