This is what eventually worked for me, but YMMV, of course. (And I'm not entirely sure every single one of these steps are necessary..)
- Switched my fitbit account to using a google account login.
- Deleted my Aria 2 scale from my fitbit account
- Removed one battery from the Aria 2
- Pressed and held the small black button above the batteries while reinserting the battery; continued to hold the button for 15 seconds.
- Stepped on the scale: it recognized me, displayed my icon & body fat%, but then showed an error when trying to sync the data to my account.
- Started the process in the fitbit app to install the scale
- Pressed and held that black button on the scale for 5 seconds
- Went through the wifi setup procedure, scale was added back into my fitbit account on the app
- Performed a weigh-in on the scale, scale recognized me, but with a new icon, waited for the scale to sync.
- After the sync, the fitbit app now showed the weigh-in.
- Went into Fitbit App->Devices->Aria 2->Users->Me and changed the icon back to what I'm used to.
...and that's it. Seems to be working now. I wish that I could say that fitbit-support <at> google dot com was of some help here, but they weren't. I just managed to find a solution by trying various random things.