I have not tried this - my worry would be that the PPU officers would inform the relevant authorities and the Americans would somehow be warned, although the PPU would probably deny this. They PPU might ask to see evidence of a visa to check you've told the truth about your conviction to the US authorities (as you would obviously have to lie on an ESTA), although I don't believe they have the absolute power to demand such evidence.
Personally I think it's best avoided until you are off the SOR. If you absolutely must go, I would advise doing it from a third country, e.g. go to France, duly do the registration for a trip to France, and get a return flight from France to the USA. But that would involve telling a lie (thus breaching your SOR requirements and risking 5 years' prison) when registering for foreign travel, answering "no" to "any other foreign countries to be visited in addition to the destination country". You could argue you hadn't intended US travel at the time of registering for your trip to France, proving this by only buying a ticket to the US after arriving in France - at short notice i.e. great cost and it would look fishy if the PPU found out.
Another way - unlawful as it would be contrary to your SOR requirements - would be to travel to Northern Ireland, cross the unmanned border to Ireland and do a return US flight from Dublin/Shannon. A further advantage of travelling from Ireland is that all flights from Ireland to the US have US pre-clearance in Ireland, i.e. you do all the immigration formalities in Ireland, then once you're on the plane you are treated as a domestic US flight and you can be at ease as there are no controls once you arrive in the US. However, even though you don't have to register to travel to Northern Ireland, if you are away for 7 days or more then you would have to register even for internal UK travel. You could legally not register for UK (NI) travel of less than 7 days, but then you would be illegally not registering travel to Ireland (risking 5 years' prison), then do a rather quick visit to the US from Ireland and return to NI and then home within the 7 days.
Basically unless you register for US travel, you would be breaking the law. And if you do register, there is every chance the PPU will inform the relevant authorities and they will know about your record when you arrive in the US.
I am not trying to provide ideas for flouting the law, as I have warned at each step that you are breaking the law and risk 5 years' prison.
I would wait until you're off the SOR, as plenty of people have simply ticked "no" on the ESTA and been allowed in, as the US has no access to the UK PNC, but if the PPU become aware of intentions to travel there they could then inform the US authorities.