If you call DL, they should be able to book JAX/AMS ---> then AMS/GVA as KL1927 as a single ticket.
You don't say if you're trying to book a paid ticket or an award ticket?
I run into this problem wnenever I do an overnite at AMS or CDG, because you can't book the continuing leg with a DL flight number - only an AF of KL flight number, so, I just call in! It should work.
Am I missing something?