Before I do this, I want to know if this would be legal in the United States without a license as testnet coins have no value and I will not be exchanging any real value coins on the exchange. Nor giving testnet coins any value at all as I will not have options to exchange to real coins.
You should retain a lawyer who can provide legal advice specific to your situation.
I don’t see any major benefit to what you are proposing. Trading on an exchange is a matter of updating database values. Displaying BTC or tBTC is a matter of updating the UI.
Testnet coins are intended to allow you to test your program’s ability to do things such as sign/broadcast transactions, detect incoming transactions, use coin control, and other wallet functions that occur on chain.