Finally got the tickets booked (See previous post if you want to know why it was so hard). It involved some analysis of the underlying problems. There are several:
(1) The airlines' websites don't allow you to book two tickets simultaneously. They also don't let you hold a reservation while booking another one (actually, they seem let you do this for domestic flights, but not international ones). So, trying to book a kids' ticket and an adult's at the same time is impossible if the full itinerary is not the same. But of course, we need to travel together.
(2) There are way too many flight options, so the websites prune the list they show you. Somehow, the fact that you are stopping over on the return trip changes the options you are presented on the outward journey. So, I never get the same set of options when booking my trip as I do when booking the kids'.
(3) Flying out of Oklahoma City just increases the number of options because of the variety of hubs that can be chosen.
So, the solution?
First, book out of Dallas. This addresses #3 and reduced #2 enough that it was possible to see the a few identical routes listed for both me and the kids.
But what about #1? I called American Airlines and had them make four bookings for the itineraries that I'd chosen out of Dallas based on getting good web fares. The problem was that even though the web listed fares around $1500, they would not get anywhere near that -- their quotes for trips involving Helsinki were about $3000. Curious, I asked about the kids' trip because their trip is simpler: Dallas to Chennai and back. They offered me tickets for $2000.
"But you're selling it on the web for $1300!," I protested.
"I have no access to the web fares," said the agent, "but let me see if I can get you children's fares on those tickets." That worked. With children's fares, each of their tickets came out to around $1400. More than the internet rate, but not ridiculously more.
I asked her to hold those tickets for me for 24 hours. Then, I got off the phone and went to the web to book our tickets. And wouldn't you know it? The carefully chosen routes were no longer listed!
Disheartened, we'd almost decided that I would simply bring the kids back with me to Helsinki and then back home, to Norman. They would get only a month with the grandparents, but there seemed to be no other decent option.
On a lark, I tried again, late in the afternoon. And the old routes were back! Perhaps the agent ended up blocking a lot of tickets when she made the kids' reservations.
Anyway, I booked the tickets online, called the agent, cross-linked the itineraries and ... hurray!
Except that we have 3 road trips to Dallas coming up. Sigh!