i just re-read your post. if the warranty states they will "repair any faulty part" that means they cover the labor too-not just the cost of the part. and if a faulty (ie. covered) part is stuck, the repair of that faulty part neccessaily entails removing it. thats part of the labor to replace it. I would argue the hell out of that and threaten court action. They just dont want to lose out on their warranty cause this repair will cost more than you paid for the premium. tough rockoes- thats the risk they took when they offered it to you.
Call them up and demand they cover it. tell them you will sue them for the cost of the repair and attorneys fees and ask for the name of their agent who will accept service of process. if and when they refuse, send them letter a letter stating the same thing. sounds like theyre playing you. they know it will cost you more in attorneys fees than the cost of the repair so they're hoping you just suck it up. It will cost them more to defend it than to pay it. so let them know you'll take them to small claims court. possibly sue them for damages for breach of a fiduciary duty or whatevers its called when an insurance company fails to cover a covered loss (do an internet search and get the correct legal terminology). Most states have small claims courts that have limits sufficient to cover your damages. they wont want to spend the time sitting in small claims court.
its a shot....