Unable to create "revolved loft"

I'm trying to create a surface that will cover the gap shown in my picture. Essentially, I would like to create a loft starting at 0 degrees as a semicircle and finishing at 90 degrees as a custom profile. Lofting doesn't accomplish this, however. I've been able to get some good-looking previews with a boundary boss/base, but every time I try to apply it it says there's self-intersecting geometry. Looking closely at the preview, I can see that it is indeed self-intersecting, but I only want the outer shell, not the whole solid.

I am unable to reproduce the preview on the boundary boss/base, but I have attached a picture of the part with the gap that needs to be filled and a picture of an iges file that shows the desired geometry. Any help is appreciated.