Sounds like you basically did good. Don't beat yourself up about forgetting how to say "right to remain silent". Not many of us get to practice that - especially when any role-play rules prohibit the JBTs from actually hauling you off to real jail.
Pardon the impertinence of the question, but after telling the police (politely) to go pound sand why did you answer their question about where you worked?
Having been there myself, I understand how much energy is expended in standing up against officiousness committed in the name of violating your rights because they think they can get away with it. It also can take being willing to take the ride and fight them in court.
Remember, if the cops actually can arrest you/want to arrest you they will do that regardless of whether you answer their questions or not. I'm not a big* advocate of poking the bear but there are times when you need to let them put up or shut up.
* for certain values of big.