iPhone 5S gets very hot and drains battery
I would connect to a fast WiFi and re-enable iCloud. The WiFi will drain the battery far less than the cellular modem for transferring 1 GB of data since it will finish much faster in most cases. Also, you might need to plug into power to let however much data you have sync.

Some big items you can turn off temporarily are:


iCloud Photo Stream
iCloud Backup
Any other large users of data from the settings app, iCloud and settings app, usage


Another thing you could try is to disable all iCloud settings, and enable them one by one. 

Once the device is caught up, you can test the device on LTE/3G data without the power cord and WiFi to see that the battery usage isn't due to downloading a lot of data from the cloud or putting that data up in the cloud.