Hi Hippy,
Yes, the HDMI cable was run out of the Entertainment slide, across the underbelly and up into the Dinette slide for the outside entertainment TV. I didn't run it - the factory did. But I've run other wires in the underbelly. In fact, recently, I ran 3 Ethernet cables from the Entertainment slide to the coat closet. Some club buddies assisted me. These things are very doable if you are persistent.
Are you sure about the practicalness of running 3 TVs from the same receiver in the sense that they all must watch the same channel at the same time? If you're okay with that, then proceed. There are also other protocols that can be used besides HDMI. You can use HDMI over Ethernet. Similar in cost depending on the brand of gear you get.
Check with monoprice.com on the HDMI splitters, cables as well as the HDMI over Ethernet gear. In the old days, I'd say another protocol to consider was doing it all at RF by using an RF modulator at the output of the receiver and run a single coax to each TV. But today's TVs have only an ATSC (digital television) tuner in them. I suppose one could buy an ATSC modulator - I've just never looked for one recently.
Running lines in the underbelly area is not difficult. Take your time, use a fish tape of fiberglass rods for wire pulling to make it lot easier.