I would buy an extra seat. We did this when we flew to Europe when DD was 22 months. It seems a shame to pay a couple of thousand dollars when you can get it for a couple of hundred, but we felt it was worth it for our sanity. We ended up with an extra seat on only one of the 4 long flights of the trip, which was great as DD was able to lie down across 2 seats to sleep, but the other 3 flights were all full.
For take-off, landing and turbulence, I had DD on my lap with an infant seat belt, and she sat in her seat the rest of the time.