How to fade LEDs switched by a 555 timer when supply is cut
If you dont want to use an MCU. I think you should give a shot at Farad super capacitors.

For your application it is hard to find a super capactior that can operate on 12 V so you have to make different combinations of capacitors (series/parallel).

If you use 5V LED strips it would be way better since there are already 1F 5V super capacitors available. Then you should make your calculations to find out how many capacitors you should use.

Have a look here and here to find the super capacitor that meets your needs.