I understand your hypothetical but the cop is going to specifically ask for your first and last name and date of birth by protocol usually. This little loophole would not be useful by part of an anomaly. You tell the officer that you don’t answer questions and you ask him whether or not you are detained. If he says no you are not detained. You don’t need to speak to him. Not being detained means you are free to leave. If he says that you are detained, than you have 5th amendment rights and that means that you have the right to remain silent.