Try Camera Zoom FX, one of my favorite camera apps and really out does the stock camera on almost any phone. I love this phone (I have the HD) and even with out the MAXX battery, I still get awesome battery life.
Now, I feel like, since you have already experienced the battery life on this phone, you're going to have a hard time using something else as your daily driver. I really don't think there is a device out there now that rivals in battery life, and I really don't think the Note 2 will be able to compete in battery life. And HTC has always had bad battery life, maybe the new one will be different, who knows.
Try the the camera app above I suggested and see what ya think.
Seems to me you might have an issue. See if you can exchange. Should not shut down when playing music in car mode, mine doesn't.
BUA+ --> Go into settings, accounts, tap on BUA+ and uncheck the sync options. If you sync and create new contacts under your Google account this feature is redundant and will eat some battery when trying to sync.
Also, again under settings, tap on "More..." under "Data Usage" and under "Mobile Networks" -->> Network Mode change from the default Global to LTE/CDMA only and that should help with battery life.
Also watch the battery stats in the settings to see whats eating your battery. Its always possible you ended up with a defective unit.