If you increase the size of the shaft would the larger bearing fit inside the input gear? Your shaft could be welded up to the larger diameter and ground, but would you have to replace the input gear to make it work? The more heat that is put into the shaft from welding, the greater the chance of ruining the shaft.
There are several ways to seld the shaft. TIG welding is comon, but spray welding is getting more popular for shaft build up.