I can help you by recommending something like the AG-801 battery.  I have one.  It was expensive, it's very small, it's very light, and it's very powerful.  
I've had mine for 3 or 4 years now.  Probably have close to 15k miles on it.  It stays in the bike year round. Even in the winter and I have never had to put it on a charger.  It doesn't have a lot of capacity, so don't let your bike sit with the lights on for 20 minutes or longer.  You'll have to bump start it or get a jump if you do.  

You'll need to build a much smaller box.