It's the unit ground pressure that determines how much sinking into surfaces will be a problem not tyre pressure. That depends on rider weight, bike & luggage weight, and weight distribution on each wheel. All things being equal, a rider & bike weighing twice as much as another rider & bike needs tyres twice as wide. It's pretty much a non-issue for light riders, especially kids - since smaller bikes ridden by lighter riders have proportionally wider tyres. As you go up the frame sizes, and the typical rider gets heavier, the tyres get proportionally narrower in relation to the load they are carrying.
At my weight of 14 stone plus bike, so a 16+ stone overall tyre load. I find 2" wide MTB tyres the only practical size for riding dirt & grass surfaces after wet weather when the ground is very soft. The OP plus bike is going to be at least 18 stone, so even worse!
With rough but firm surfaces, tyre sizing is just a question of rider comfort and safe loading limits, not getting bogged down. I doubt the OP will be able to cope with soft mud on that bike at their weight irrespective of tyres. Compacted stone & gravel it will cope fine with so long as the tyres are tough enough.