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The Generac 6340, like many 30 amp, 240 volt devices is designed to accommodate conductors ranging from 10-8 AWG. By staying within this tolerance, the installer is able ensure that a solid connection is made without jeopardizing the products construction. While cases have arisen where a 6 AWG conductor have been squeezed into one of these boxes, it is certainly not advised. Keep in mind that the density of these wires makes them very rigid and it is often very difficult to tuck them into the box while maintain a stress free connection n the device.
While it would be convenient to use the material that you currently have in place, it sounds that there may be complications with the install and finished fit. We would recommend that you consider replacing that 6/3 wire.