Personally if I were the bandits, the attack would have started with a booby trap/deadfall to stop the carriage, not a volley of arrows. Chasing a carriage on foot without some means to hobble it would not be a sound plan.
Given I had at least one archer, I would have them pelt the now stopped carriage, perhaps with fire arrows to flush them out into the open where more arrows would be followed up by a ground attack.
The Volley is problematic as you mention The Archer - who, unless he is superhuman, is not going to be able unleash a volley of arrows.
Being able to fight at all within the confines of a carriage light enough to be managed by a single horse seems nearly impossible.
Maybe with daggers and martial arts, but swords - I don't think so. I would expect the interior of a single horse cart to be as cramped as the backseat of a car.
Friendly fire with swords would be almost assured.
You get a little more leeway if you have a larger vehicle drawn by a larger team. E.g. Stagecoach. Even then, it is extremely close quarters. No one is going to be able to simply slip past anyone unless they have been drawn out of the vehicle and left several feet of space. Perhaps if Meng was in one of those blade-to-blade struggles which took all of his attention...
There is a scene in Crouching Tiger Hidden Dragon Sword of Destiny where there is a battle about a carriage, but virtually none of the combat occurs inside. Its just too d**n small. I don't have a you tube link to it sadly.
If you want a battle with a vehicle serving as a mini fortress of sorts, a boat might work better.