If you're after a NQY base then Air Southwest would indeed be the way to go. I don't think they have any aircraft based at EXT. Flybe on the other hand do have aircraft based there.
That being said, don't expect to be based at either of these airports if you are successful in an application to ASW or FB - The general rule is that you go where the company needs you to go. If you apply to an airline expecting to be based somewhere then you will be disappointed if they base you 500 miles away. Be prepared to go where the job sends you and you'll never be disappointed - you'll probably move up the career ladder considerably quicker as well.