To upgrade the Batmobile, you need to use the in-game portal as Batman, choose the Upgrade option, then select your vehicle.
The Model Color Chooser is in the bottom row, in the eighth column. You will need to unlock the third Batmobile model, as well as all of the prior color options, before you can unlock the color chooser. If you don't have the model unlocked, follow the guides for this trophy to unlock it:
Once you've purchased the Model Color Chooser, select the Batmobile model (top left), then go to the color chooser and cycle it forward with
until pink is selected.
I don't believe I had to save it to the toy tag, but if it doesn't pop, try doing so by pressing
.