Small UPS for bluetooth audio device
Why not put a USB battery in between the device and the USB port?

Just make sure you get one that supports "pass-though" like this $10 one..

https://www.amazon.com/Patriot-Ultra-Compact-Rechargeable-Warranty-Smartphone/dp/B00DW9PUD8/ref=as_sl_pc_ss_til?tag=joshcom-20&amp;linkCode=w01&amp;linkId=S5RBTYAWFPEL3FXT&amp;creativeASIN=B00DW9PUD8

"Pass-though" means you can have the battery connected to both the power supply and the device at the same time and it will "pass-though" the power. Once the power supply is removed (car turned off), then the battery will kick in and supply to the device.