Thank you very much to all those who replied.

Armed with information provided by all of you, I was finally able to make the booking after some back-and-forth with KLM's customer service rep on FB and also on the phone. (They really don't make it easy, do they?)

The rep on FB confirmed that as Elite members of another skyteam airline, I am entitled to select a standard economy seat free of charge. However, there's no way for me to do this on my own on the KLM website. They offered to help me make the booking on their end but I didn't feel comfortable sending all the personal information to them on FB messenger. So I called KLM's customer service line in the U.S. (by the way I couldn't even find this number on KLM's website, I had to google it). It appears the customer service is run by Delta in the US. After some trial-and-error, the rep on the phone was finally able to complete the booking for me and let me select seats without additional charge.

A few minutes later I received the confirmation email not from KLM, but from Delta (with Delta's logo and everything). I then went on KLM's website to make sure that I am able to retrieve the booking record there, and I did find it.

Again, many thanks for all your help!!