Yesterday, I received an I-797, Notice of Action, from the Department of Homeland Security. This informed me that my application to have the provisional status removed from my green card has been accepted for adjudication. This process will take at least 330 days, so my green card has been automatically extended for one year, with work and travel authorised.
Basically, this gets me the right to stay in the US past 20th March, when my green card would originally have expired. With that, another part of our grand masterplan falls into place.