My definition of a cycle with sidecar is "two utterly different vehicles with one license plate". Left turns and right turns are two completely different types of maneuvers.
Let's say we put the sidecar on the right-hand side of the bike, since that's what I had.
You now have good protection against tipping to the right, regardless of the weight in the sidecar.
You may or may not have much protection against tipping to the left. If the sidecar is light, you have little protection. If the sidecar is heavy, you have good protection -- but you have to drag all that @#$%^&* weight around with you.
In addition, the heavy sidecar will drag the vehicle toward the right even when you want to go straight.
So the heavy sidecar makes it easy to steer the vehicle toward the right when you wish to turn in that direction.
But the heavy sidecar makes it difficult to turn toward the left. In fact, with the motorcycle, my primary means of turning left was to apply the rear brake and let the momentum of the sidecar push the vehicle around toward the left. This is not practical with pedal power, where you have barely enough power to get around at all, much less waste momentum by applying a brake just to steer.
Conversely with turning toward the right. I used to apply engine power. Again, precious little surplus power available with leg-power.
I loved my sidecar. For several years I installed it in the autumn and took it off in the spring. All winter I traveled mostly sideways. (Exaggerating just a tiny bit.) On steel-studded tires. This was 40-plus years ago in rural Norway -- on nice snowy icy roads, with thick cushy snowbanks to crash into. As tried to hint at... great sport.
But as Playa transportation for a person with a weak leg? Fuggeddaboutit.
As for hauling 240 quarts of ice and beer around BRC in a pedal-powered sidecar.... I once filled my sidecar with firewood. Piled high. Had to make all my left turns by making three right turns.
A sidecar is an abomination of engineering. It's only excuse for existing was to allow transportation of more people and goods with a motorcycle, in a world short of automobiles and money to buy such.
But hey... prove me wrong, and I'll help celebrate with all that beer!