Or, if you don't have a welding rig, maybe make a sturdy shelf out of whatever material you have on hand. Ideally, sturdy sheet metal with a 90 degree bend to bolt to the hand truck's uprights.
Or plywood with braces.
Then tie or bolt the new shelf firmly to the hand truck's shelf and uprights.
I think the challenge would be to make the attachment more secure and solid than the original "set" of a big pot on a small shelf. Spot-welding rebar plus a long steel plate WOULD be very secure!
Hmm, how about come kind of sturdy "sleeve" or "cup" with an open front, that would be a good fit for most of your problematic pots? What to make it from would depend on what you have available. It might have to "hug" a range of pot sizes, and that might not be practical.
You would roll the pot into the "cup" and tie it in firmly. Then run the handtruck under them both and tie them to the handtruck uprights.
The "cup" would have a shelf long enough to extend well past the center of gravity of any big pot.
It would either have some kind of wrap-around sides, or a back with eyes for ropes and bungee cords. Or straps arranged like a straitjacket!
The back would have some means to quickly tie the "sleeve" or "cup" to the hand truck. Short lengths of rope?
Maybe that is more elaborate then necessary. If ropes or straps could be permanently attached to the handtruck's uprights, would that be enough to hold large pots in place? Or would that depend on the shape of each pot?