You're always going to struggle with a 1-5/16" socket.

The nut is actually 1-1/8"... and you'll need a deep socket.

Google for Amrstrong tools and the size you want.  You should be directed to a few of their products in different drive sizes, 6 point, and 12 point.  Their data sheets include the dimensions of the socket.

I am in the process of installing the drive gear and nut and found that my 1 5/16” socket 3/4 drive will not fit inside the recess of the drive gear to properly torque the nut (2A3548). I am wondering if the same size socket in a 1/2” drive might have a smaller OD on the socket to allow it to fit inside the recess? I’ve already checked sears, ace, autozone, oriellys but no luck finding this socket in 1/2” drive to check. Any suggestions or experience? Thanks.