Widening load bearing door 4" & retaining existing header
That 2 x 12 is over engineered.  I'm guessing this is an older house, built when large dimensional lumber was much cheaper.  

First, See this question Proper size of header to support new door in load bearing wall?

It looks like you can conservatively get away with 2 2x6's on edge for the header, and some cripple studs to fill the empty space (Ignore text on img) - 

I'm still trying to figure out how you're getting four inches in there without moving that wall stud at the far right of the picture.