I guess we are indeed in uncharted territory here, and the banks are in an unenviable position. It might make sense for them to block a sale if the seller is playing games and selling below market price. For a bona fide seller selling at current market price, I'm not sure that it makes sense for them to block the sale, even if it leaves them with a lump of unsecured debt. The alternative to not selling is just increasing arrears and higher debt with no prospect of it being repaid (sounds like the Tony Soprano school of lending).
I doubt if there are many people in the OP's position who could find someone else willing to secure their negative equity. And is it really fair for the banks to look for such security - they were happy to lend on the basis of the mortgage, so why should they look for a second bite of the cherry now?