I'd say you’ll need to pay $549.. No way to prove to them that the drop didn’t cause the charging problem even though it was having the issue before.. A case on such an expensive phone is a smart insurance policy...
There likely isn't a way for them to know whether or not the drop happened before or after you talked to them about the issue. There's what you say and your word, but that would come down on them believing that or not, having nothing else to go on basically.
Try but don’t be surprised if you send them the phone and they return it to you and tell you they can’t do anything with the phone because it’s has exterior damage.. They do it all the time, many threads about Apple refusing to touch the phone because of broken/cracked displays, rear glass, water damage... As I said you could of avoided all of this by using a case, you’re fault completely for the situation your in. No one on this forum can authorize the replacement, that’s Apple’s call and you’re at their mercy..