Currently you can't.
From the documentation:
The module is based on the CoreBluetooth framework, but doesn’t expose all of its functionality – for example, it only implements the central role, i.e. you can connect to peripherals but not have the device act as a peripheral itself.
Background audio and background Bluetooth are both valuable to have in specific use cases. But they can lead to battery life problems easily with programming mistakes. Would it be possible to provide per file activation? That would at least show that people are actively seeking this capability and are not going to blame Apple for any extra power drain.
Personally I would find background Bluetooth quite useful to log some data over longer periods.