Have been through it before. You gotta get used to the thought that the bike fate is sealed.
Regardless of your good intentions, the fact is that the bike is not yours, that you don't know the owner and that he has the right to do anything he wants with his bike, included and not restricted to let it rot until the city takes away the remains.
I had a similar experience to this, and believe me, when you ask the owner about the bike, he will suddenly recover interest.
If the bike is rotting and you "save" it, you are just saving yourself some hundred bucks that would cost you a bike in similar shape, with title. I'm the kind of people that, even after having "saved" the bike, would never feel it really belonged to me because I knew how I got it.
Face it, we can't save all the bikes in the world, some of them must be junked. After all, that's what gives more value to the ones remaining. I would not feel guilty to take some parts before it is scraped, but even then, for the price that those parts sell I could just buy them, at least I would keep my peace of mind.
Of all the advice read here, I would take showing at 00:01 of the deadline. The city has given the owner the chance to recover the bike, and you have saved the city the expense of removing and scraping the remains. That's the only "fair" way I see to make that bike "yours", at least physically.