This is awesome- congrats. KLM is a great airline- I think you'll really enjoy it. Air France and KLM are sort of sister airlines, known as Flying Blue awards program. Just putting my experience on your radar, but hopefully you don't end up with this issue. I booked economy seats on AF/KLM through Virgin Atlantic, because as you saw, they can be much cheaper. However, maybe four months prior to departure, AF ended up canceling the flight. Because AF/KLM only allocate a certain number of award flights to VS and none were available near my original date, my only option was to take a cancellation, rather than switch to another flight. So then I was stuck rebooking them directly with AF, but the economy rates went way way up. So I paid double or triple AND I had points stranded with VS.
I also had award tickets for that same original flight booked in biz (for different passengers- some were in biz and some economy) directly through Air France. Because I had booked directly with AF, I was able to switch to any AF/KLM flight of my choosing, when they canceled my original flight.
So sometimes it pays to book directly with the airline program you are flying. But for that price, I would have probably jumped on that deal as well!!