It’s a little late for me, so the quality is poor, but this might be a start…
I think that just adding a filleted section would be problematic because the part you would run nearly coplaner to the part that is there.
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So, the part underneath needed to be removed, or really the affected face needs to be rebuilt.
What I did was:
I copied everything.
I exploded the finial.
I created 2 curves by copying the side edges of the finial section I moved out.
Using your height reference line, I split the lines I created, using the point option.
I curve-matched the curves of the lines you intended to be the fillet edges, to the copied edge curves.
I joined the fillet curves to the edge curves.
I used EdgeSrf to create the long surface from the result edge curves.
(Sweeping and lofting likely won’t work.)
For each side, I used the remaining curves to edge curve the end cap.
There still may be gotchas when assembling it, but this may work, though you have to be more careful than I am, while trying to stay awake.
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Also, it looks like in the photo, that the main sides of this thing look kind of like a pointy oval.
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If you have one of these swoopy faces, you can also make it wider, and then array polar, and then use them to trim each other and make a solid. Surfaces can overlap while creating a solid, as long as the result would be airtight, and it’s cool to watch.
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You can also make a subtraction tool using one of these swoopy faces and a few more surfaces, and then subtract the tool from the post, or whathave you, and keep rotating it between subtractions, or perhaps just do a boolean 2 objects using the surface and the solid post.