Increasing my phone's charging rate while tethered
Charging rate is set by hardware and software, and can't be modified by software only. 

Charging rate depends on the following: 


resistance between D+ and D- (determines is it a official, unofficial or USB charging)
hardware technology and limitations (QuickCharge 2.0, 3.0)
cable length and quality (bad cables usually have more resistance)


For the S3, as far as I know, you can charge slow (USB, 5V 500mA) or fast (wall charger, 5V 900mA). 

The charging type is set depending on the resistance, which is different for wall chargers (fixed to activate fast charging functionality in phones chip) and standard USB charging. If you want to charge quickly while tethering, the only valid option is to have your phone connected to an official Samsung wall charger which will ensure a higher charging current. There may be an option to set the USB connection to "Charge only" but that doesn't guarantee that the fast charging will be turned on.