I use Sleep Cycle as an alarm clock/sleep tracker, but it doesn't offer a bedtime reminder functionality. Is there any way to use the native Bedtime alarm on iOS 10 without using it as a wakeup alarm as well?
It doesn't seem like there's a way to do this natively. This is probably because the Bedtime feature is a supercharged alarm -- one which tracks your sleep (although it's not as advanced as Sleep Cycle) and shows you the bedtime notifications. Being able to remove the alarm notification completely would void this core functionality.
However, you could use some other app other than the Clock app for the reminder functionality.
You could simply use the Reminders app, which allows you to set daily recurring reminders. You could even set it to give you a "bedtime reminder" every weekday with a custom repeat option. The advantage of this is that it's completely native. From the notification, you can snooze the reminder by an hour (unfortunately the 15 minute snooze option was removed in iOS 10). A big advantage is that it will stay on the screen even if you unlock your phone, until you mark it as complete. This does not seem to be the case with the clock app's reminder. However, this can be a disadvantage as well, if you want it to dismiss automatically.
Another option would be to use an app such as Due, $4.99. It allows you to set a recurring reminder, but also allows you to snooze it from the lock screen. I haven't personally tried it, but it might allow you to customize the snooze duration. However, I wouldn't use the repeating reminder option, as it seems to have become broken due to iOS 10's new notification API.