If I understand your question correctly, then no. The Google/Apple API allows apps that use it to access Bluetooth in the background, without being the front-most app and without the phone being awake. Those requirements do exist for apps that are not using the Apple/Google API: the app has to be actually awake and running to function, hence the power drain.
The NHSX app operates the same as confirmed by Apple engineers; they allowed the workaround NHSX came up with to achieve the same outcome.