Is it possible to run an iOS 7 device as a Bluetooth LE peripheral (iBeacon) and have it advertise in the background? I have been able to get it to advertise in the foregro
No, iOS devices only advertise iBeacon when the app that does the advertising runs in the foreground. so, if you switch to another app or if the device goes to sleep, the advertisement stops.
Of course, if you really want the advertisement to continue, disable the idle timer and do Guided Access so that the iOs device does not go to sleep and no one can switch to another app.