You're avoiding the inverter issue...and you apparently don't realize that the cigarette lighter outlets are fused at either 10A or 15A.
As someone who has owned multiple inverters, who has wired up...

1. If you had looked, you would have noticed that many of these cushions already have time and temperature controls, e.g.:
https://www.ebay.com/itm/281289985853

2. Your build uses a heating pad...

Everyone here is making this difficult.

Buy a heated seat cushion for your car, the kind that plugs into the 12V outlet. This should cost you under $20 on ebay. You want the kind that only covers...