It turns out that the BECM (body ECU) wakes up from sleep if it hears anything coming in from the RF keyfob receiver, on 433.92MHz, a standard ISM frequency. If it's not a valid fob code (and there is a surprisingly complex rolling code sequence) it'll go back to sleep after 20 minutes or so.
The oil tank gauge for my central heating had a remote sensor that... yup. Transmitted on 433.920MHz, of course.
The aerial for the keyfob is on the rear right corner of the vehicle, the oil tank for the heating was on the left-hand side of the driveway right at the back of the house, so when I backed into my driveway it was right beside the receiver.
I managed to do just fine with popping the lid off the oil tank once a month and looking inside to see how much fuel was left, having taken the battery out of the tank sender.