Dave, thanks for getting us started. I'll continue on from where you were, using the same assumptions, Kyocera 135 and Morningstar SS-20. I'm going to give broad figures, there are always variables that I can't get into here.
You'll want a breaker or fuse between the module and the charge controller. Either using the math or the spec sheet. isc X 1.56 to give us 8.73 X 1.56 = 13.6amps, round up to the nearest available of 15A. Or use the Max Series Fuse listed in the specs of, drumroll, 15A.
Coming out of the charge controller into the battery, you want another one. If it is the 20A charge controller, 20A x 1.25 = 25A. Not sure if they make one at 25A, may need to round up to 30A.
Now the second system, that's tricky without knowing what the model module is. Let's keep it simple with Evergreen 210W 12V modules. Since you are using the 12V inverter, I am assuming the modules are in parallel to keep at 12V, so you would have a combiner box with a 20A breaker on each output of the module (11.23Isc x 1.56). The output of the breakers would be combined. http://www.altestore.com/store/Enclosures-Electrical-Safety/Electrical-Enclosures/Combiner-Pass-Through-Boxes/Midnite-Solar-MNPV3-Combiner-Box/p4567/
Assuming Morningstar TriStar 45A PWM charge controller, 45A x 1.25 = 56.25A, round up to 60A breaker.
Between the inverter and the battery, 300W can work as low as 10V, 300W / 10V = 30A potential x 1.25 = 37.5A, round up to 40A.
Now I was making a lot of assumptions, why don't you give us a call here and we can go through it with facts.
Solar Thermal Queen