Battery drain with higher volume
No. Bluetooth is a digital data connection, and it's power does not vary based on the raw amplitude of the audio signal being digitized and transmitted. What would change the battery life  is how much data is being sent, and a volume difference is not that. Additionally,  Bluetooth audio allows for volume control outside of the raw audio signal,  and that also doesn't increase power usage.

Any given audio stream will use the same amount of power as the next really. Bluetooth is not a very power hungry spec.