Can an electric stove infinite switch for a small burner be used to control a large burner?
I spent some time researching these infinite switches because they are failing on our 25-year-old GE range but by a different mode than you have. Ours are losing the lower end. They don't heat at all on low but do work above a certain setting.

The infinite switches are different part numbers for the two different burners and it is not clear to me that it is "safe" to use an infinite switch for a small burner on a large one. The wires and the sensing element in the small infinite switch might not be sized to carry the larger current drawn by the large burner and might overheat or fail somehow.

The current through the switch is determined by the size of the burner. The infinite switch regulates the heat of the burner by square wave time modulation of the current on a time scale of maybe a second. So when the current is on it would be at a higher value (9 A?) on the large burner than on the smaller one. It is questionable in my mind that it is wise to use the infinite switch for the smaller burner on the larger one.

I suggest you get the specified one for the larger burner and get a new heating element for the smaller burner.