You have to keep in mind that they aren't necessarily going to tell you what the actual complaint is. It could be anyone - a neighbor, enemy, friend of the child, busybody, anyone making an unfounded anonymous complaint. They have to follow up, and they have no idea in the beginning whether it's specious or a situation requiring immediate action.
We're all outraged, but it's important that you show no anger nor drag your feet on any part of the process. Don't escalate unless it's absolutely necessary. Provide them with references who would be familiar with your situation and who you know will not torpedo you. If you can put them in touch with your fiance that can probably end it, but let her know ahead of time. Document all contacts and conversations.
Not having a car can only be part of the allegations. It sounds like she's leaving you a way out, because she suspects that the allegations are bogus. I think it's probably a good idea to just buy a beater and satisfy them. Keep the priorities straight: resolving the situation is first and foremost, after that worry about fighting back. Confronting them head-on by filing complaints may not be the best way to extricate oneself - how does that actually help you? If it's not something like a personal vendetta against you that's not going to knock out the investigation. Cooperate, compromise, and explore further options when it's all over.