Yes, you need to setup the phone as new. I understand your issues on the whole process of setting everything from scratch, I'm afraid that's the only way out. However, if you look closely, on the whole, it boils down to following issues only.
- Contacts - You will get this back once you setup your iCloud account on the phone so it shouldn't be a great issue.
- Apps - Do a transfer purchase to your mac/pc before going for firmware complete reset, once you set the phone as new, you can always sync back the apps. This beats the inapp things like game levels, or your pdf docs synced earlier. iBooks can be restored, otherwise enable document and files backup to iCloud in settings before resetting the phone, then you can get everything back from iCloud.
- Mails - If you have a mac you can keep all your mail accounts configured on Mail.app, post install you can bring them back from Mail.app to phone easily; otherwise, I'm afraid you need to do everything manually
- Settings - you will have to setup your wifi configuration, VPN if any manually from the scratch
Dispite all the hasseles listed above, the battery fix would be a hot sun on the face in a winter day. Go for it!
