As said by CHMultimedia, Background restriction won't consume mobile data in the background and it is not a problem to connect the Bluetooth devices.
Please make sure the Bluetooth device is within the range and in the pairing mode.
Go to Settings >> Connections >> Phone visibility >> Activate it.
When connecting it, make sure the devices are within 5-10 feet of each other to get a better experience.
Then Perform clear cache and clear data for the Bluetooth.
Go to settings>> Apps>> Bluetooth>> Clear cache and clear data.