For what it's worth, I don't think that Harry's back is explicitly tied to his gig as the Winter Knight. I think the bout of paralysis he experienced in Cold Days was a result of him (briefly) reneging on his Deal with Mab.
First, obviously, Harry's paralysis is taken care of before his term as Winter Knight starts, and that service (as I understand it) begins before he has the Mantle; killing Lloyd Slate is actually the first command she gives him as Queen to Knight. Second, he's agreeing to service explicitly in exchange for healing him (and, you know, buffing him up with Winter Juice).
All of that is to say that by saying "Screw Winter Law," he tacitly rejected the terms of his Deal, which yanked his non-paralysis away temporarily.
I agree with your overall point, but I disagree in some of the details. IMO Harry is agreeing to serve Mab for the healing and for time to rescue his daughter "before his service begins." Meaning that he gets access to the perks of the WK (the healed back and power) before Mab can give his orders so that he can rescue his daughter. The killing of Slate was necessary to free up the power so it could be passed from Mab to Harry.
"While Lloyd Slate lives and breathes, he is my Knight," said Mab's voice. "Take Medea's bodkin, wizard. Take his life's blood."
I stood there holding the knife and looking at Lloyd Slate. The last time I'd heard him speak, he had begged me to kill him. I didn't think he'd be capable of even that much now.
"If you would be my Knight, then this is the first death I desire of you," Mab said, her voice almost gentle. She faced me across the Stone Table. "Send his power back to me. And I will render it unto thee."
It wasn't so much an order from the Winter Queen to the Winter Knight. Instead it was a necessary step that Harry had to take if he wanted that power. He was completely free to back out and pursue one of his other options if he so desired.