Here's a theoretical solution, but admittedly there is still a lot to work out...
The characters themselves will basically have to balloons. As far as finding a material that would suit the finish of the character and be able to stretch like a balloon...that's another story. But for all intents and purposes the characters are balloons.
Each is stationary, sitting on a cactus. That means you don't really have to worry about operating the characters (except the growing and shrinking). That also means you don't have to worry so much about weight.
You could use water instead of air. So, if the large character (at the beginning of the story), filled fat with water, sits at a point on the cactus that is higher than the smaller character (who may be filled with just a little water) and the two characters/balloons are connected by a tube...the water will find an equalizing on its own. It would be similar to how water towers provide water pressure at your sink or shower. You could regulate the transfer of water either by the size of the tube or by way of a valve of some sort inserted in the tube between the two balloon characters. You may try looking around at what is available in the aquarium section at pet store as far as valves go. A valve would allow for the bursts of growth and shrinkage.
It may also be possible, assuming the cactus is hollow, to build a mech that would pull the mouth (where the tube connects) of the larger balloon into the cactus as the water is transferred, thus keeping the "skin" tight. The converse could be done with the small character, feeding more balloon up as the character grows.
Again, still a whole lot to work out. The idea is certainly a tough order.