Extrude a Brep along Curve while Rotating????

I have made a Brep that moves along a curve and rotates as it goes, now I am trying to find a way to "capture" the form that the Brep would make if each instance of movement were combined...

By using the Record component and animating the slider (Right Click Slider and Change Frame Count for Higher "resolution") I am able to create many Breps that start to define the form that I am looking for, but I think there is a better way that I am missing.

Ideally, I would be able to get a single BREP that is a simplified version of the combination of each instance along the curve.

I am unsure as to how to deal with hundred or even thousands of BREPs that just slightly overlap-- the BREP Union component just gives me a warning when I try to use it in this example.

Any thoughts are appreciated.

Thanks.