Light box
Well, Q_Hybrid, I just spent quite a bit of time composing a response and it got lost in the ether during submission, so this may be a bit more terse.
Calculate your electricity costs for operating the installation directly off the line. I'll bet it will be only a few cents per day (unless the installation is in Las Vegas). Amortization of the cost of a battery and charger system will way exceed the cost of operation of a few CCFLs off the line. And don't forget that someone has to pay for the cost of electricity to charge the batteries unless you use solar panels (another large cost to amortize). Additionally, the charge-discharge efficiency of the battery system knocks off another 20% or so of the energy you pay for (depending on the battery chemestry you use). So don't expect to save money by using a battery system instead of line-power.
I'm not ignoring the possible advantages - fun, aesthetics, safety, ect., of using batteries, but avoiding the cost of electricity should not be expected to be one of the advantages.
Nevertheless, to estimate battery capacity, take the current draw of the inverter you want to use, multiply by the number of inverters, multiply by the number of hours of operation per charging cycle, divide by the efficiency of the charge-discharge cycle of the battery technology you use (maybe 80%), multiply by an arbitrary factor to take into consideration battery deterioration and less than perfect cells (maybe 1.25), throw in a safety factor so your installation doesn't grind to a dim halt during display, and you have the minimum amp-hour capacity required of your battery. All this for a battery voltage compatible with your inverters. If you want to use a voltage regulator to maintain constant light power over the entire day, divide by the estimated efficiency of your regulator - maybe 85 to 90 % for a high quality DC-DC converter, much less for a linear regulator (depending upon the current and voltage drop across the regulator). Throw in the time required for someone to manage the power supply to your installation each day, battery replacement due to over-discharge when someone forgets to turn the installation off or overcharging when someone forgets to terminate the charge, and the line-powered option starts looking pretty attractive.
Don't forget that the small 2 or 3 mm. diameter CCFLs put out much less light than even a small line-powered compact fluorescent from the hardware store. After all, their primary application is back-lighting for LCD screens, not room lighting. You might want to try out a single lamp and inverter before investing in a complete system that you might not find to be satisfactory.
As a wild guess, I would expect that you would need about a 10 to 15 amp-hour battery at 6 volts to operate four CCFLs for a day. But don't take my word for it. Run the numbers for your equipment choices.
I don't understand the comment about the cats-eye LED cluster. Remember, however, that CCFLs are significantly more efficient in lumens per watt than LEDs. (This is difficult to compare directly because data is either not readily available or the two types of sources use different types of specifications due to the directionality/spectral/application differences.) This is not to say LEDs aren't well suited to your installation, just that you will get fewer lumens per watt in out of LEDs.
If you use matched inverter-tube combinations, you don't need a ballast. The inverter IS the ballast, i.e., it controls and limits the current through the tube.
The main point is to have fun!
awright