Well it seems ultimate combat and ultimate equipment are in conflict.
But looking into this it shows exactly the same propulsion requirements for all three types of wagon in Ultimate Combat despite the fact that a light wagon is 10 x 10 and a heavy is 10 x 20 (literally double the size).
I believe it's pretty obvious that Ultimate Combat is in error here.
Also, Ultimate Equipment was published after Ultimate Combat. I know for conflicts with the Bestiary the general precedent was to use the more recent publication.
I see no problem with your general idea of placing a chariot on a larger vehicle to carry your group and the chariot. And only attach the chariot for combat.
However, your mount simply cannot pull a heavy wagon by itself. If you can find a small enough vehicle that still accommodates the size you desire and only requires one horse then you're alright.
the second printing of UC still has the error in it. The heavy wagon has the same team of animals the medium wagon does, which is a big giveaway that its wrong.
as an aside, if you have all these wagons, and two chariots, why not just tow one of them, rather than carry it?