There is already a codeshare as far as I’m aware. Both KL and BE flight numbers are available on connections.
What strikes me as odd is how SOU can’t sustain this route given that one would have thought the majority of customers would be connecting. It always annoyed me how high the fares were for connecting flights. If they don’t compete more effectively with LHR and LGW on connecting air fares then it doesn’t stand a chance.