I'm arriving later still at SFO in two weeks time and faced the same issue, except room rates are less than half that and I'm headed in the opposite direction the next morning.
My recommendation would be to pick up a car and get over the bridge as far as Berkeley that evening. This is because a) the last hotel shuttle will likely be at 00:30, so you might end up taking Uber to the hotel; b)I would find hanging around for a shuttle, or an Uber, at that time of night particularly annoying; and c) you will face the morning rush into SF to reach the Bay Bridge on the following morning. If you pick up a car and drive to Berkeley, the drive will likely be 20 minutes faster at midnight than at 7am, and you will have saved some 30+ minutes messing around with hotel shuttles.