If you join BAEC, you will NOT be credited for the flights you have already deposited into your Alaska Air account. There is no double dipping. However, you may be able to get the miles transfered from Alaska to BA. It may take a lot of effort but that is your only option to get BA mileage credit.

Yes, BA does keep track of this. I had deposited BA miles into my American acct and called about getting credit for BA. I wanted to switch between the accts. The CS rep looked up my info and saw that I had already received credit for the miles and told me BA credit would not be allowed. When I asked about moving the miles, the CS said she wasn't sure how to go about it. I did not pursue the matter further.