Buy a seat! I've done the mainland (TX) --> Hawaii route with my son 2, most recently when he was 9 months old (crawling, not walking). Thank goodness for the seat! Once he was asleep I could walk, stretch, use the restroom, and eat. Additionally he was happy and comfortable in his carseat. He did some crawling around on the floor but was also happy to be strapped in. Once he was able to walk flying was more challenging since he wanted to be up and exploring. The seat is best since it is more likely that the child will sleep better and you'll get a break from the very exhausting task of entertaining a child on the plane. A seat of her own will also give your daughter that much more floor space for playing.
Happy travels!