I have not recreated the model to test it, but I'll guess that the sweep fails when it tries to go around the sharp corners of the path or guide. Sweeps don't change direction very well. Think of it like driving in a car, you can't corner like that, neither can the sweep.
You might try putting generous sketch fillets on the path and guide sketches to see if it works as a test.
Why not just extrude a triangle, and cut a smaller triangle through it? Two extrusion beat a sweep in terms of complexity.
If you had to, you could extrude both profiles at once from a single sketch with two closed profiles.
Thanks for responding! I can't extrude it because the rectangle that I have here is just a representation of a more complicated shape. Let's pretend that rectangle was a hexagon instead. I would like the hexagon to sweep, becoming thicker (larger radius) at the opposite end of this triangle, and thinner where the profile is defined. All the sweep combinations I've tried haven't allowed me to do anything like this! I'm simply trying to do a non-uniform sweep :-)
Thank you though, if it's not possible, then it's not possible.
Maybe try a loft? You could setup the starting hexagon, and the ending hexagon and loft between the two.
It may be possible to get all three sides at once, but one loft per side may make for a better model.