I don't want to manually switch bluetooth on and off periodically.
Is there any way (including apps) that can enable bluetooth when the device is charging, but disable it when the charge cord is disconnected?
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No, There is no way to switch bluetooth automatically since there is NO API let you do that except jailbreak. I don't think enabled bluetooth all the time will drain the battery life so quickly, bluetooth is not the root reason.
You can follow this
Guide to Solving iOS Battery Drain