I recently ran into a problem when using the Pillow app (and the same issue happened with Sleep Cycle as well) in conjunction with Bedtime on my iPhone. What was happening was that Sleep Analysis in the Health app would only display a solid chunk of "In Bed" from whenever I put down my iPhone until the time that I responded to the Bedtime alarm the next morning.
Essentially, the data from Bedtime was overwriting any of the useful data from Pillow. This was very frustrating, because I want to see the more detailed, useful information about my sleep without having to look at the dedicated 3rd-party app every time. Those statistics are, after all, the entire reason for using an app such as Pillow or Sleep Cycle in the first place.
It seemed like I would have to choose between using Bedtime and using a sleep tracker. This, too, was frustrating because I enjoy aspects of each. Bedtime allows me to set a regular schedule, complete with reminders and the Do Not Disturb During Bedtime feature. Plus, the alarm sounds are way better!
I finally figured it out, thanks in large part to a post by Michael Kummer.
As it turns out, you can re-order the priority of different data sources in the Health app. Open the Health app on your iPhone, go to Health Data > Sleep Analysis > Data Sources & Access. Tap Edit in the top right. Rearrange the data sources to the desired priority. Tap Done.
Wow. It seems like this should be way more obvious. I still cannot find any documentation from Apple that explains how to do this. Maybe it's obvious for others and I should be embarrassed that it took me so long to solve, but I'm just happy I can finally enjoy the best of Bedtime + Pillow.
Hopefully this post helps someone else experiencing the same frustration!