wmeek, you ask if a change in the K factor would change the voltage drop much. Well the voltage drop is directly proportional to the 'K' factor; double K and you double voltage drop. Of course, the 'K' factor is pretty much a property of the material in question, however it is strongly affected by temperature.
The K factor for copper at 20C is about 10.2. If you are increasing the size of the wire so dramatically in order to deal with voltage drop, then the wire won't self heat very much, and will pretty much sit at ambient temperature. Cooler wire has lower resistance, thus producing less voltage drop.
While you may be limited to 120V, is it possible that your sign circuit could be subdivided? If you can break your single 120V 13A load into several 120V sub-loads, then you may be able to run a multi-wire branch circuit, and effectively convert the sign into a 208 or 240V load.
In terms of how much voltage drop you should design with (3%, 5%, more), that will depend upon the specific design and requirements of the LED power supply in the sign. It is possible that the power supply is some sort of switching regulator that will happily function and maintain proper output with a very large voltage drop. On the other hand, if the sign uses a simple resistive current limiter, then a reduction in supply voltage could make the sign much dimmer. Without knowing the details of the load, there is no way of knowing what sort of voltage drop is acceptable.
-Jon
