I have not found a way to do this.
What is your purpose . Can you explain more.
I suggest you to use a balloon that linked to your part , and use configuration name as a text . Also you can add a general table that linked to custom properties (like configuration name ) , then when you change the configuration view the text (in balloon ) and cell in general table will be changed automatically . Unfortunately when you insert a BOM table you have to choose configuration of inserted part (not configuration view )
See the attached video.
capture-1.avi 9.2 MB
Not exactly how it should work in Solidworks, but a nice way around Solidworks lacking this capability. Very nicely explained as well!!