It is all very well meant behavior to think of composting and it does work where you have a garden space, a farm or a municipal composting program. But when it comes to being out in the wilderness areas that need to be kept in their own natural condition the rule pack out what you pack in is the rule of the land because that is how it exactly how it needs to be done. You are not out there homesteading, you are just passing through as a visitor. You have a moral responsibility and it means that you do not change that natural eco-system you are visiting with your personal garbage of whatever type it is you have brought into the area.
You don't put your food waste into the ground in a forest because you would have to disturb the natural ground covering to do so. The food waste might also contain viable seeds from plants which do not belong there. It might attract wildlife which would dig the area back up to get at the food it smells.
Realize that you will not have sufficient time to fully compost the food waste you produce. That you can't have a fully closed composting system so it needs to be kept outside of your living space. If you kept it inside you would have a major issue with flies and maggots in your dwelling area. Fermenting food is like mana from heaven for all kinds of flies, a prime food source and nursery to lay eggs in.
If you keep it outside then you will also be attracting rats, mice, coyotes, racoons, bears, etc to come to your vehicle. The rats and mice will get into the wiring on your engine and eat it. All your food including pet food should be stored in airtight containers.. Garbage should also be in an airtight container for the same reasons.
So basically you are going to be very unhappy with the visitors who are going to be attracted to your compost bucket system.