It's a cable shifter, which is very effective in isolating the hand lever from the gearbox. You wouldn't expect too much vibration to come through from the gearbox.
However, the fact that you can't feel it through the steering wheel, leads away from a TV problem and back to mounts. You said you can feel it through the pedals, if it's the floor then it increases the evidence for a mount.
It's really impossible to diagnose over the web, it's a case of take it in somewhere or start changing things. Mounts are a sensible place to start, even if they aren't the issue, replacing a worn one will make the car feel better regardless.