iPhone Battery

Why questions can't be answered here. This is a user to user only forum. All we can tell you is Apple has always refused service on devices which have been tampered with by anyone other than Apple or an Apple Authorized Service Provider. Your phone still works, it will just always always show that it has inauthentic parts installed. The ONLY place you can get an authentic battery put in your iPhone is Apple or an Apple Authorized Service Provider. If they are telling you they would be willing to let you pay them to put an authentic battery in your iPhone, I'd take advantage of that. Often, they simply won't touch a device which wasn't serviced by them.

That sounds about right. Apple requires an exchange for a battery service regardless of whether or not the existing battery is Apple OEM (installed officially) or otherwise.

I hope they do for you. They don't have to, so you'll be fortunate here.