I just caught that I misread your original post so I am going to correct what I should have said.
Apple's recycling partner I assume is the 3rd-party company you mentioned you were referred to. It is true that Apple is not involved in the transaction if you look at the fine print buried when you do the trade-in, and the deal is between you and that company, not Apple. It's unclear but it sounds like the damage happened between you shipping to them? If so they presumably took pictures of how they received your trade-in as I believe that is standard for them. I don't know if that will help you with FedEx or not.
But all in all this sounds like a very tough and I suspect un-winnable situation. I would imagine both the recycler and FedEx are going to blame everyone else (you and the other party) which makes this difficult for anyone to accept responsibility since the recycler can blame fedex or you (saying it was already broken) and fedex can say the same thing that it was you, or that the recycler accepted deliver without noting any physical damage.
This is just awful all the way around my friend.