I didn't see food, clothing or anything else in the trailer...so plan on being at the GVWR from the start plus some stuff in the truck bed.
Those little munchkins...they get bigger, they get bikes, they take friends, you take firewood, etc... A basic 1/2 ton 4 door truck is a nice option for a family of 4.
When buying, probably the biggest number to watch is the Payload sticker by the door. 1/2 ton trucks have a wide variety of payloads depending on how they are equipped. Avoid the weak ones with 1400-1600 payload (a family of 4 with teenage kids can easily eat up 700lb of the payload just with bodies) and try to get something up in the 1800-2000lb range. That will give you a lot more flexibility in the future if you get a bigger trailer.
The turbo powered engines they are putting in 1/2 ton trucks really come into their own in mountain driving. A Naturally Aspirated engine loses around 3-4% of power for every 1000ft of elevation...so if you are up at 6-7,000ft you are down by around 20-30% on power. Now a truck that was just a bit of a dog at sea level is really struggling. With a Turbo, it force feeds air into the engine largely negating the altitude based power losses.