Chances are, I should beat my 10 yr old, for sitting here and accessing my AC account a few hours ago. Butt whipping pending....
But I guess it's my fault, for teaching her technical things.
As phones get more sophisticated, of course they demand more power. This holds true when charging, especially when using the phone at the same time.
This is probably why your phone has to be plugged into the car charger, when docked.
The same as when you plug into a USB connection (pc), you'll notice the device gets hotter than normal.
Will it get hot enough to crack the screen, no. In most cases, it won't. Typically, if it were a bad battery, the battery would, well, become distorted, and in some cases, explode. Unlike "my daughter" said, with enough rdx to blow your hand off.
But a blown battery, could cause damage, as shown in a lot of cases.
A bad battery, could, and in most cases, cause a lot of heat when charging. Could it cause your screen to crack, well yes. If the screen had flaws to begin with. But it would have to get damn hot.
Can you prove it to your carrier, probably not. But I would guess, you can negotiate, and come to some sort of agreement. It may not be the best, but with the circumstances, you should be able to meet halfway.
Remember, they don't want to lose you as a customer. Just stand your ground.