OP, I've been through this but I'm no expert. My father died two years ago, and I've managed to forget most of what I had to learn during the almost 10 years he spent in 'care'.
Still, if your father's savings have been used up, then I think the LA will have to pay.
Most homes have to keep in mind LA rates, although they may pretend that they won't accept them, because most of them have to rely on the LA to pay the bills. As others have said, you can't be forced to top them up. Also, not many families can afford to pay privately (i.e. with no LA contribution).
I think you need to speak to your father's Social Worker about this. Unless your father is in a particularly expensive home then the reality is probably that they're simply asking for what they expect to receive, and the LA may simply have to stump up and pay it. Could be the LA are paying below the going rate ATM.
As I've said below, I have every sympathy because I know how traumatic the prospect of a move can be. My father was on the brink of having to move from the place he'd been in for a number of years (not because of finance, but because we'd had to complain about the standard of care and so they'd kicked us out). He managed to die of pneumonia in the space of the week preceding the move, and so didn't have to go through it. I hope for better for your family 