That's an interesting scenario! I hope it’s not too late to weigh in, I apologize if it is.
I think it's possible that Gollum would have gone into hiding with the Ring, and I'll get to that scenario in a bit. But to start by going off on a tangent, I wonder if we aren't meant to think that Gollum would have tried to claim the Ring and use it (its main power of domination, not the invisibility power). He used to be hiding with it, but he had gone through a rough time since then, and it seems he wanted revenge. He says to himself about Shelob: "And when we’ve got it safe, then She’ll know it, O yes, then we’ll pay Her back, my precious. Then we’ll pay everyone back!" It sounds like he wanted to use the Ring to get back at everyone (presumably Shelob, Sauron, Sam, Aragorn, Thranduil and his elves etc).
If he had tried this, or maybe even if he had just claimed the Ring without using it, I think it would've led to Sauron recovering it. In one of his letters (246), Tolkien wrote that if Gollum's redemption hadn't been interrupted, his love for Frodo would have enabled him to realize that he "could not use the Ring and had not the strength or stature to keep it in Sauron's despite." He also writes that Sauron would have swiftly reclaimed the Ring from Frodo if Frodo hadn't been attacked by Gollum (unless Frodo had enough sanity left to cast himself into the fire with the Ring).
So I think the most likely outcome would be that Sauron would take the Ring from Gollum.
Anyway, back to the scenario where Gollum disappears with the Ring.
I could be wrong but I always got the impression that a successful defense against Sauron without destroying the Ring wasn't possible, although some characters initially hoped that it would be. In the what-if scenario, nothing really happens that would improve the military situation compared to what existed in the book at the time of the Last Debate. The Rohirrim arriving, Aragorn coming on the ships, the Witch-king being defeated, winning the battle of Pelennor Fields, all this stuff happened in the book. But Gandalf still says that victory by arms isn't possible at this point, that’s why he advocates the "feint" idea (which is very risky and not that likely to succeed, so not something you'd argue for if you had other options). He says: "Victory cannot be achieved by arms, whether you sit here to endure siege after siege, or march out to be overwhelmed beyond the River." So he says that even a defensive posture couldn't save Gondor and he certainly doesn't recommend an offensive. Osgiliath used to be a large city and had been heavily fortified. I don't think it would've been possible to retake it while outnumbered 1:10 by the "defenders" (Sauron's forces).
I think without the destruction of the Ring, Sauron would have won in the North. The fate of the Gondorians (the ones who didn’t fall in battle) would have probably been the escape into the mountains scenario predicted by Beregond and Denethor, but they would’ve been hunted down eventually. I guess my impression is that what we see in LotR was meant to be this "all or nothing" situation where just prolonging the stalemate further wasn't possible anymore. I might be overlooking something though.
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