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.
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.
Also, it looks like in the photo, that the main sides of this thing look kind of like a pointy oval.
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.
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.