Apparently this a restriction that AA has placed on awards using another airlines miles. They have a rule that "You cannot redeem partner airline miles with your partner frequent flyer number and obtain AAdvantage program benefits such as priority boarding and access to preferred seats". They appear to have taken steps last year to enforce this at least on Avios award tickets by not allowing updates to the FF#. That explains why the Finnair trick works for most but not everyone.
When you book on BAEC and select yourself as the traveller from their pre populated list they automatically enter the BAEC number and you are stuck with that unless you can find a friendly AA agent to override the system, which is becoming harder and harder. There was a workaround which was to book flights as an "other traveller" which allowed you to enter whichever frequent flyer number you want. But it seems that household accounts do not allow bookings for anyone not already on the friends and family list, and I even see current reports of not being allowed to this anymore without a household account. Gotta love all these "improvements" that airlines are making these days.