So what you want to do is book flights for your trip via AA.com but keep your AA number for your flights but use BAEC for the rest of the family so their miles post to BA?
Yes that's possible. When you add each family member to the booking remember to add their BA number in the appropriate place in the booking. IIRC there is a specific question about FF programme and number after inputting their name, passport etc
Then when you want to use the BA avios in the HHA you would then book the flights via your BA account and then utilise the avios across all members pro rata to the balances.
When the OP says 'premier' I assume they just mean get status and not have *proper* premier status