What do you think of my plan?
You've thought it through very well. It should work just fine.
Instead of an inverter, you might want to check around and see if you can get a 12V adapter for your laptop. That's a less messy installation because you just have one cord to connect the laptop to the cigarette lighter instead of a cord and brick to connect to the inverter and another cord from the inverter to the cigarette lighter. You can get a 12V adapter for most laptops.
...ken...