I think you could, and I'll explain, but first a disclaimer: don't try this, because it's a good way to fall resulting in damage to your FJR and injury to yourself.
The AE does engage the clutch with the engine off, if it has been off for a few seconds. This is how you can 'park' an AE on a hill and not worry about it rolling. The clutch doesn't disengage until the starter is pressed. I know this from trying to push a bike that was in gear, with the ignition turned on, but not started.
I think if you had the bike on, shifted it into 2nd, got it rolling down a hill, and then turned the ignition off, it would engage the clutch after a short while. The trick would be having the bike lurch as the clutch engaged, being able to control it, and then reach with one hand to turn the ignition key back on... all without crashing. If I was taller I'd run an experiment, but as it is I can barely get my toes down. This is something I would keep in mind for a desperate situation, but wouldn't do otherwise.