The flight from SEA to LGB arrives at 1:10 PM so for the return boarding should begin by 1:20 - 1:25 PM if everything is running smoothly. Check flight #300 before you arrive to see if it is arriving at LGB on time. You will probably have no problem checking luggage into Seattle except a possible wait for your luggage in Seattle.
However, as an unrelated note at LGB: I don't trust the America West staff with my incoming luggage because they somehow manage to turn just a few feet distance from the plane to conveyor belt into a 1/2 hour or longer baggage delay. Always a different excuse: belt breaks down, no staff is available, airport management refuses to respond to the belt's mechanical problems because they only respond when the airline (AS) contacts them directly not passengers.
As a heads up I saw the America West check-in agent being admonished for staffing the AS gate after the last call for PAX. She defended herself to her co-worker by saying that AS PAX can still get on stand-by for the next flight but the other employee corrected her saying that AS does not allow same day stand-by without paying the $100 change fee (sic) and she should shut down. immediately.