At 12 feet deep plus 1' of overhang so call it 14', half the roof is bearing on the back wall, half on the front, so 7' x 32' opening= 224 square feet bearing on the beam. 4000/224=18 pounds per square foot. I use 25 snow+ 10-15 psf dead load here in the VA mountains. I think you are underestimating the load. I believe Anchorage uses 50 psf but you need to get good local numbers.
You would possibly be better looking at a gabled truss spanning the opening. Is the building overall going to be longer than 32'?