It won't be NFC as that requires much closer ranges. Typical remote frequencies are 315, 434, 868 MHz etc for the remote. NFC is really 13.56MHz. 125 kHz are the older prox systems. BMW Comfort access (keyless entry and start) is 125 kHz apparently.

It's unlikely the key fob will be transmitting regularly or the battery would not last long at all.

The car would be sending a signal to the fob (when you touch the door handle to unlock or press the start button). The fob receives that and then replies appropriately to authenticate the request.

So you'd need to be able to generate something that the fob will reply to and then detect when it does reply.

I don't like your chances.