InsideTracker does not yet support sleep tracking compatibility with Apple Watch outside of tracking Sleep Duration and Time in Bed. This extends to other apps which send sleep data through Apple Health.
Apple only recently launched Sleep tracking with watchOS 9 (previously only newer series Apple Watches would track sleep duration). Now you can see a breakdown of how much time you spent in each sleep stage in your Health app if you choose to sleep while wearing your Apple Watch.
If you're seeing the sleep breakdown available in your health app between REM, Core, and Deep sleep, you should not expect to see it in your InsideTracker app. We're currently working on improving our capabilities to accommodate Apple's latest release.