I may have a solution for extreme battery usage, but not unexpected shutdown.
Simply switch the data from LTE to 2G, reboot the phone and switch back to LTE.
Before doing this,
I made a test by remove the SIM, put on air plane mode with Wi-Fi & Bluetooth on.
I made a week.
But didn't use the phone at all only reed some email couple of time.
No unexpected shutdown between 30-40% (It happen once a while).
After 5 day, the battery pass from 29% to 1% instantly during I was reading email.
Then I check couple of time to see when it will die without abuse of the phone except check if it's still on.
I made 5th, 6th and 7th day on 1%.
Then I was hurry to see when it will shut the 7th day,
I was going crazy using my iPhone 4 that apple "iOS update" kill it to a toooooo slooooooo phone!
I check some email and it finally die.
Then I put back the SIM card but switch off the Data.
I made almost a day, but an unexpected shut down at 42%.
Then I switch back the data, but it was still drain the battery as crazy.
Then I remember that in the past I had an battery drainage, and I found a trick on the web that it is simply switch the data from LTE to 2G, reboot the phone and switch back to LTE.
I do this this morning, and after 6h of use (reed couple of email and take 10 picture, I'm at 89% now.
But I also deactivate Siri, push email on some personal email and couple of minor thing to save battery usage.
Yesterday, I download the Battery Life app and I have 21.1% Battery Wear Level. (2300/2915 mAh)
I'll keep you posted if the battery still draining fast.
But I don't think it will help the 30-40% unexpected shut down or the 29-1% &1% for couple of hour/day.