Why not just get a direct train from Southampton Airport Parkway (Station) to Bournemouth station (about £11 from memory, but haven’t done it since January so not certain on that!) and then catch a bus from the station to the Bournemouth Travel Interchange, and then you swap to a bus there which runs directly to the airport…just a thought! Try this for bus directions…
http://www.flybournemouth.com/airport-information/bybus.php