So here's what I'm thinking. Use your computer during the day and keep your solar panel hooked up to it. Hopefully it's sunny

then your battery charges and then near sundown, you bring the battery inside and hook up your inverter to it. Most have jumper cable style clamps so it takes no wiring and like 5 seconds. Turn it on and plug your computer into its AC outlet then boot it up. It'll run off power from the battery via the inverter. Start mining and hope that your battery has enough juice to not drop below approximately 10.4V before morning. Then wake up, make delicious eggs, read slashdot, brush your teeth, then go shut down your computer. Then unhook the inverter and put the battery back outside on the solar panel and repeat daily. If you find your battery running out, calculate the draw and overall capacity of the battery via one of many methods and then add 1 or more additional batteries in parallel to have enough power.
Sounds a bit too many manual operations involved. Why don't you consider hooking things up in parallel, or triple source a cheaper UPS so that it would run off AC, Solar or Battery depending on which is available?
Alternatively, wiring the solar/battery directly to the most demanding +12V (with a regulator of course) auxilliary power inputs of the graphic cards so the solar powered stuff will just drive about 80% of the cards requirements and theoretically easier to expand to more cards simply by adding another set of panel-batter "modules"?
After all, 3 sets of 120W panel + battery each capable of handling 3KWh is going to be cheaper and more reliable since a single failure doesn't take out your entire setup