I'm having trouble understanding exactly what you are trying to do. Maybe a diagram would help.

My deck is cantilevered three feet in two directions.

I had to use two LVL beams. There's a 3.5inch wide one that runs straight out from the door. And a 5.25inch one that runs diagonally from under the door to the corner nearest the camera. The company that supplied the beams had a structural engineer that signed off on the design to the satisfaction of the local building inspector. It took several iterations to come up with something that they would approve. The diagonal beam made the framing a bit tricky but it sounds like you won't need to do that.

Anyway - the LVL beams let me do something that my contractor thought was not possible!

I don't see an issue. It's no different from framing a screen porch with a perimeter beam and then hanging the attached deck joists off that. What does your permit authority say? At the end of the day, they're the ones you need to please.