Bike options for winter commuting in the mountains
Here are the things I believe you should be considering when making the decision.  

Fat bikes climb just fine.  Yes, a road bike will climb a paved hill more efficiently than a mountain bike under the same conditions.  But, a mountain bike will also climb a muddy single track that a road bike would simply fail to move at all on.  Efficiency and utility are trade offs in this respect.  

A cyclocross bike will be more efficient in many conditions.  However, at some point your cross bike will become unrideable (too much/deep/wet snow) and a fat bike will plod on.  Depending on your weather and commitment to riding, a couple days a year where you shelve your cross bike and take other transportation may be acceptable for you.  However, if you want to make sure you can ride as often as possible, a fat bike will offer you more commuting days.  A fat bike also offers an advantage in this area as far as not finding out halfway into your commute that you are going to be pushing your cross bike the last 3 miles.  It's slower, but a bit more honey badger.

As far as safety, I'd much prefer to descend with ANY snow on a fat bike than a cross bike.  Thinner tires may cut through snow for traction, but when descending that sinking has a tendency to yank the front wheel all over the place.  A wider front tire will float and may have a tendency to wash, but not dig and turn.  

A fat bike would open up new options to you for winter trail riding.  A cross bike just doesn't even enter into this equation.  However, if you aren't into trail riding, it isn't really an advantage to you.  Personally I enjoy winter trail riding above all other forms of riding and I live in the right place for it, so fat bikes are THE bike for me.