score:13
The US customs declaration form explicitly prompts visitors to declare only those items that will remain in the US. You therefore do not need to declare the ring.
If they search you, you'll ask them to do it in private (away from your girlfriend) and tell them that you're giving it to her and she's going to take it back to the UK. You won't be able to prove that, but they're unlikely to suspect you of lying. They're even more unlikely to search you in the first place: such searches are very rare.
It might be a good idea to bring the receipt showing how much you paid for the ring, in case things go very poorly and they decide to charge duty, but that possibility seems so remote that you are probably going to be fine if you don't bring it.
The receipt will also be useful in the (also unlikely) event you're challenged about the ring on return to the UK.