Another thing that can help in general with thickening is to form your object into a solid and then shell it rather than trying to thicken a surface. That's because when thickening a surface it's easy for the "side wall" direction to not be very well defined, while with a solid the shelling the side walls get to come from existing geometry.
But you've got enough problematic things in this shape that just that alone isn't going to get the job done I'm sure.
- Michael
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