How, specifically, does an electric meter measure wattage?

Of course it's possible, but how and how safe it is may be a problem. What kind of power meter is it? What is the supply input to it. This usually runs in big numbers: 50A, 100A, 200A?

If it's any of those numbers, you have a big problem outrunning it without blowing something up. Not only would you have to use extremely large wire (or run the risk of the magic smoke escaping) and some pretty sophisticated protective clothing. You're going to be channeling huge quantities of energy that can easily vaporize a few inches of copper, starting fires in insulation rated for 400C.

Sucking it faster than the meter can sense is not a good idea.