Are you positive that your ducts are connected? You don't want to have to go past the heat strips (if you have electric).
Is it possible to go under the house and through the baseboard? Or through the attic and behind the crown molding (if you have any)? I've done it this way many times. Just be careful taking the molding down and you should be fine (no major structural damage). In fact, an ethernet port is a good thing in a modern house.
