I am going to say no, you can't.
Since you hold a NM license as a state resident that is your "primary" license- that is the standard which law enforcement is going to hold you to. I do not think it is safe to assume otherwise. If you only held the FL one, you may have a better chance of making the argument.
If you want to be able to carry a larger caliber, why not just go through the process to do so and have your NM license updated?