It's actually not IME *too* difficult to get miles lifted and recredited.

I recently had a flight incorrectly post to my AA account (switched FF numbers at check-in, had my BA EC number on the BP, etc) - sent an email to AA to ask them to remove it and credit it to my BA a/c, received confirmation by email within 2 days and a snail mail letter a week later confirming that AA had removed the flight and requested credit from BA.

Still waiting for the BA credit though . Certainly worth joining BA Exec Club to get a membership number and then calling/mailing Alaskan to ask them to do the same.

Good luck.