Its the actual feather producing that edge blur.
If you need a rounded rectangle, use the Rounded Rectangle Tool, with settings for corner radius, fill, etc: https://helpx.adobe.com/photoshop/how-to/edit-rounded-rectangles.html
It's not that difficult, start duplicating the same character you already have:
Delete the hidden shapes
Move, scale, transform or flip horizontally the shapes that remain in the side view
Add the new shapes that will be seen in this view
Add shadows and details
I think I'd use the InkTan extension to generate the outline. Still needs a bit of path joining afterwards, but it will be exact by default.
To turn these three objects into a single one, connect the ends of the tangents with new segments, so they form a rectangle, then do Path > Union with all three objects.
Group all objects and make a copy. Lock the original
Select the new group and merge it into a single object with the Pathfinder Merge tool (either Window > Pathfinder, top left icon OR Effect > Pathfinder > Merge)
Set the fill of the new object to white and place below the original object.
Scale the object up with the frame handles while holding Option + ...
To recreate this effect, I usually duplicate all of the artwork (or specific layers, that form the outer shape), Insert the Stroke I want and put it behind the original artwork.
That will create the effect you want but you remain a dirty vector, so, as an extra step I use the pathfinder tool to merge shapes together willing to create one single shape.
The case is already solved with a tangent drawing extension. Actually in this special case the tangents can be drawn also without that extension. They are as long as the line between the circle midpoints and they can be made by duplicating that connection line. Placing them properly with snapping happens when one draws a 90 degrees rotated copy of the line. ...
Illustrator hasn't an one click transformation which solves your problem. But if you make a high resolution, say 2000 x 2000 pixels or more sized raster image copy of your shape you can make a polar to rectangular distortion filtering in Photoshop (ADD: check, if that is included to Illustrator's own collection "Photoshop's effects")
In Photoshop make an ...
Make sure the circles are aligned either vertically or horizontally, then draw a rectangle between them, ensuring that the size matches and that it ends at the centers of both circles. Then merge with the pathfinder tool and rotate/recolor as needed.
When you draw two overlaping circles and a line, a simple shield shape (similar to a gothic arc) emerges in the overlap:
If you move the circles closer to each other, you'll get a narrower shield:
If you instead move the line upwards, you can extend the circle segments on the sides:
If you combine three circles, you can get a circle segment at the top:
You will need to group the object, but the gradient tool for some reason does not allow you to apply a gradient to a grouped object directly (as you can see me demonstrate in the middle of the attached GIF). I'm not sure if this is a bug or not - but there is a workaround!
Group the text and the object, then add a new fill from the appearance panel.