Why is the generator turned back off for starting the second engine on a King Air?
Yes it loads up the battery, but the battery is sized to do this. The generator is not necessarily sized to do this, especially on the ground, so this procedure gets you to take the generator out of the system for each start.

On an Alternator, like on your car, there is a phenomenon known as 'forward-stripping', which is where there is such a demand for current out of the generator that it overloads the rectifiers (that convert the created AC electricity into DC for output). So here, during/after the engine start, there may be a large charge request by the battery which might forward-strip the gen's rectifiers.