IF it’s a view (looking NW) from the West Wing Press Briefing Room, it might be the building (row house) just west of the Rochambeau statute in Lafayette Square.
That might be the Peter Parker House, 716 Jackson PL NW (intersection of Pennsylvania Ave NW and Jackson Place NW.)
This view below, is from the NE corner of the intersection looking back at the West Wing. 2 green canopies can be seen on the WH lawn that likely protect the press from inclement weather when broadcasting.
One end of a green canopy can be seen out the window, behind the camera man, and beyond the canopy, the Parker House.
If one drags the image about 90 degrees clockwise, the Parker House will come into view.
I don’t know.
I don’t think the brick building with the awnings would show up so “big” in the windows based on the distance it is likely from the West Wing judging from this aerial view: