Here are the results from my google searching on the exercise bike:
Which has ALL manner of wacky bikes that use your upper body. Neat animation half-way down the page. I don't see any that combine standard bike pedaling and upper-body, though. But a lot of rowing bikes.
And the gold mine:
People have been doing this for a LONG time, it seems. I still haven't seen anything exactly like my vision, but plenty of things that come close- mostly 2-wheel, interestingly. And some really entertaining stuff.
Last but not least, here is the exact thing I was thinking of (in a bike format, but the same idea) - there are also other patents from the page above that are trikes, with more or less the same thing - double handles that serve as extra propulsion and steering: