By your description I would just stick with the iPhone 5. If we're talking about benchmarks the iPhone 5, S3/Note 2 are about the same. In most cases the Note 2 wins, and then the iPhone 5 and S3 are about even. The S3 being compared is the one with the Exynos 4412(?) quad core, so the American variants probably don't stand as much of a chance against the iPhone 5.
The S3 has a decent battery life, probably in the top 10 of the longest battery life in phones currently, but I don't its good as an iPhone's battery life. I just got my hands on a CDMA iPhone 4 with a bad esn, and I was amazed at the battery life. 100% with Airplane mode on after a good 8 hours of sleep. I let kids play minecraft on the iPhone 4 for a hour and it only went down 7-8%, while my S3 would have gone down to a lower percentage.
It's hard to choose an Android phone over an iPhone if you're looking for a phone with a great battery life and little bugs. While all phones have their own bugs, what I noticed (IMO) is that the bugs occur due to the apps itself. The developer will update an app and it will kill the experience. I have about 10 apps right now that I am not updating due to the many bugs that are occurring on that version.
I chose the S3 because of the difference of design and size compared to an iPhone. Although the looks of a phone is not everything and may not be important to everyone, it was a big decision. Most phones have Dual-Core SoC's, this had 2GB of RAM, and I liked the design.
I personally get almost 1 Day and 12 hours of battery time with my phone usage.