To upgrade the Batmobile, you need to use the in-game portal as Batman, choose the Upgrade option, then select your vehicle.
At the top of the screen are the three Models. The first is unlocked by default; you need to upgrade more abilities on the Batmobile before you can unlock #2, and then repeat for #3.
You will require a total of 325,000 studs (minimum, depending on your upgrade choices) and 10 gold bricks to unlock the third Model.
This trophy pops once you've unlocked the third Model, and written it to a Toy Tag.