Can I hang two inch blinds with only one inch of available window sill overhang?
I have 2" blinds from Lowes, probably the same one. Our windows had enough over hang though.  If I recall, there are little metal rectangle boxes that get mounted / screwed into the top of the window, and then the the blinds slide into the little metal box to hold it in place.  These little metal brackets have numerous holes in them on both the side, and on the top.  I think there were four screw holes, one in each corner on the top, and two on the side.  If this is indeed the type of bracket your blinds come with, then you would most likely be able to get three screws in place (1 side and 2 in the top).  Then repeat that with the other side of the blind.

This should be more than sufficient to hold it in place.  I think it would look a little funny though, I would probably go with a 1.5" for looks.  But you could use a valance to cover it up and no one would even know.