IPhone 4s only charges when plugged in and turned off?

I recently found that my IPhone 4s only charges when the phone is off when plugged in. But the phone shows no sign that it is charging, no red battery when its turned off, i just turn the phone on and its charged to 100%. I've had the phone for longer then a year, so i took it back to the company which sold me the device and they told me they had not seen or heard anything like this, They told me to take it down to my local apple store. But after booking an appointment apple told that they haven't heard of this before either and they think it could be internal apps crashing or the software, they done a test on the battery and they said the battery is fully healthy and the dock is not dirty or broken. They also said the problem is not the fault of mine. But because i have going over the years warranty apple have said it will cost £126 to be sent to be repaired and the company which i bought off have told me it will cost £146 to be replaced which i can't really afford at this moment. I can't connect the iPhone to the computer or anything. If any body has any idea of what this is or what i can do to fix it will be much appreciated.

I would say the battery is bad on it, so get a new phone