If weight is an Issue, I would take a G1X and a small tripod. You'll have more important stuff to take than just a camera. First Aid, A Good Knife, Food, Utensils, Flash-lights, sleeping bad and tent or hammock, Fire kit, ETC. You want to be as light as possible.
If Weight is not an issue, Take whatever you want.
When hiking Mt. Fuji, he won't need any of that stuff as there is a lodge near the top of the mountain which serves curry and has sleeping bags for everyone. There is also numerous checkpoints and a boatload of people do the climb so it's pretty safe.
The only thing here that might be good is a head lamp or a flashlight. And if so, i'd go with a LED one as it is a long hike and the batteries will last and if you are slow you'll reach the lodge after nightfall.
Personally, I would just bring the wide-angle lens to cut down on weight. Mt. Fuji is not an easy climb if you've never been hiking before. Also, you might have trouble setting up a tripod at the top as it will be swamped with people.
One hint, if you have another camera ready, you can have one takign a shot of the sunrise, and another taking a shot of Mt. Fuji's shadow stressing out from teh base of the Mt.. It ends up being close to a perfect triangle and is pretty excellent looking.