Can I keep a copy of my drivers' license in the car so I don't have to take my wallet when I go to the gym?

Sure if you want to.

And if your car is broken into or stolen, you will be at a large risk for identity theft.

There are many wallets for exercisers to store an ID and other needs. Here is some examples:

Amazon.com : RunWallet, Grantwood Technology's Workout Wallet, MILITARY GREEN : Ankle Wallet : Electronics

You can, it is not a valid ID and you will still get a ticket for not having your license with you. At least in Georgia, Can not speak for all the states. A copy of your license is not enough and if the police officer stopping you, is a "hard " he can write a ticket. When as a police officer, you did not have license with you, we run your name and birthdate and it shows us, you have a license, but you still get a ticket for not having license with you.

You can put your real license in the glove box, while you are in the gym. Or put your wallet in the trunk.

But a copy may be OK for a few police officers, but if it will not satisfy most.

I wasn't thinking of a COPY being used as valid ID.

If you leave copies of your SSN and bank account numbers too you can make it a little easier for the people that steal your car to get everything else you have.