I want to make sure I do not exceed the 200mA max of the Arduino.
Isn't that oft-quoted number the max output of the Arduino CHIP? Since you're not driving the darlington arrays directly from the chip, it doesn't matter. (OTOH, there's a board maximum current as well, which is less obviously defined (it depends on the input voltage.)
The ULN2803 has a very simple internal circuit; each driver has a 2.7k resistor leading to the base of a darlingtonton transistor, so the current is just (5-2*Vbe
Yes, having a separate power supply for the shift registers should solve any problem.
1.35mA per output is (significantly) less than your LED mockup is drawing, isn't it?