It seems you can't reorganize the configuration in that way.
Not sure some API can do the trick.
You can reorganize the parent/child configuration structure through a design table. I can't remember how off the top of my head but you can look into that.
Make a design table as suggested.
Then edit it by selecting "Edit table in new window.
Save (or copy-paste) to a clean Excel workbook.
Delete the design table and ALL CONFIGURATIONS from the part (you will have to make one temporary configuration though).
Edit the Excel workbook so the parent-child relations are as you want them. Might need some trial and error.
Insert a new design table in your part and select "From file".
Delete your temporary configuration.
Any assembly or drawing using the part will recognize the new configuration as long as it has EXACTLY the same name. Even it is in fact a new configuration and is placed differently. But only if the original configuration has been completely deleted (not renamed)
At least has this method worked for me.
Remember to back-up your part.
OK, I guess that works, because you deleted all parent configurations.