All three of my now-middle school age kids have been flying since before they were born and were in an airplane within a month of being born. At 19 months, your daughter is old enough to learn to keep the headset on. Just make it non-negotiable, like being in a car seat. I found it quite easy to train my kids to wear them. If you don't have a kid-sized headset, that might be part of the problem, as they not only will fit a small head better but also don't clamp as tight.
If she absolutely will not cooperate and you are willing to give in, I don't see the harm in occasional flight with no hearing protection. I learned to fly in the days before headsets, and it's not really all that bad in a light single -- unless you do it all the time.
