I have never experienced short battery life with the Kodak Z-1012, and I use the standard Kodak KLIC battery and the Chinese equivalent. I always get around 250 shot on a single battery charge.
The Sony H-20 as I mentioned previously has just about the best indoor flash range and exposure in the compact zoom market niche. I also does very well out of doors and I am attaching a personal photo sample.
Also keep in mind that the Sony H-20 is physically smaller in size than the Kodak Z-1012.
Have a great day.