I would think that instead of big with a gas guzzling engine you might go for something more like the old Willys Jeep. Fairly small, compact, with a small, simple, fuel efficient engine that carries a lot of weight for its size through gearing. The old Jeep only had, depending on year, about between 20 and 60HP worth of gasoline engine with the majority or wartime production having 60HP. At capacity, considerably more than 3/4 ton, on a flat road it would only do 40 mph but, as the saying goes, it could climb a tree. Small, a tight turn radius, and narrow helps when you need to drive around obstacles and on sidewalks.

I would go for a small 1/2 ton diesel truck outfitted with a 3/4 ton rear axle. Diesel saves a lot of complication by eliminating the ignition system and diesels have become much lighter than they used to be. Something with gearing making it just capable of interstate highway speeds. Most highways are legal down to 45 or 50 mph around here but I would want at least 60 mph at half load so I can take it out on the highway before any disaster.

In an emergency the 'need for speed' is limited. After an earthquake you don't want to go flying off a broken overpass and keeping your speed at around 25 mph should increase your mileage considerably. Which further saves weight and trouble. Something with a frame. A lightweight one ton winch, rigging straps and a couple of snatch blocks should allow you to shift anything you need to.

A small trailer would add weight and save wear on the suspension. It would also allow you to bring a larger load but leave a majority behind so you can scoot up ahead and check things out. A small teardrop trailer would give you a place to sleep and keep your gear dry and relatively secure.