Another point – if you let your TIP expire, my understanding is that you may be barred from bring another vehicle into Mexico (or trying to return with that same vehicle), because you did not abide by the rules of the TIP.
If the TIP is expired and you want to keep the vehicle, you can get “safe passage” paperwork which gives you some number of days to remove the vehicle from Mexico without it being seized along the way. Otherwise, you’re just taking your chances driving around in an illegal vehicle which could be taken from you. If you removed it via “safe passage” at least you could still sell it for whatever it’s worth in the US.

Mexperience is good at answering most questions, but I’ll give you my take in the meantime…
If you got a TIP, the “free zone” is irrelevant to your situation. The TIP allows you to drive your foreign-plated vehicle anywhere in Mexico for the entire time the TIP is valid. If you have a temporary residency card, the TIP expiration is tied to the expiration date of your residency card. Period. You MUST remove the vehicle from Mexico by that date. Even if you renew your temporary residency, the TIP is not renewable. You must physically take the vehicle out of Mexico and if you want, you can bring it back by getting a new TIP, which would now expire when your new residency card expires. Those are the rules you agreed to by applying for a TIP.
The officer you talked to is incorrect to say there is no free zone there, but what he might have meant was the only way you can legally enter Q.Roo under the “free zone” rules is to enter Mexico from Belize. If you enter from the US without a TIP and drive to Q.Roo, by definition you have driven illegally through Mexico to arrive at that “free zone”. (But that would not be the case for you, since you say you got a TIP.) Is it expired? If your TIP is expired and the vehicle is still in Mexico, then yes, your vehicle is subject to seizure/impound. I don’t think it would matter that you are in a “free zone” of Q.Roo. (Maybe don’t show them an expired TIP…?)

As far as I know, you cannot register a foreign-plated car. You would have to go through the regularization process to import it. (Many people think this is not worth the time and expense and just buy another car in Mexico.) But I would think that if you bought a mexican car and are using it in Mexico, you would be required to register it in Mexico.
Hope that is somewhat helpful.