The chassis is mounted on a sturdy board that slides into slot in the cabinet wall. I wouldn't be too worried about the chassis flopping around or causing havoc while the set lies on its face. It's worth reaching in to make sure the tubes are securely seated.
The bell of the CRT is its heaviest part; that would be lying against the curved bezel on the front of the cabinet. Not the worst way for that piece to travel.
The yoke is suspended from the top of the cabinet with a bracket. If the set is on its face, that somewhat heavy piece will be hanging in a direction they never intended. If it was loose for some reason, there might be some danger of it working sideways and putting force on the fragile CRT neck. Before I moved the set face down, I'd at least reach up and feel that thing to make sure it's securely mounted without a lot of wiggle room.
I'd remove the knobs if you transport it face down. No reason not to.
My biggest concern would be the combined weight of everything mashing down on the front of the cabinet, which wasn't intended to support the TV's full weight. I have transported heavy TVs face down without problems. Each one's a little different. I would definitely remove the two knobs from the front doors before trying this.
An Outback wagon seems like a tight fit. I don't think I could get one in and out of an Outback without two helpers: one guy in the back and one on each side in the front (reaching through the back doors) to provide lift on the corners while you slide it in.
I doubt it weighs more than 150 pounds. We just moved my CT-100, which the manufacturer gives as 175 pounds. That TV feels quite a bit heavier than the RA-113.
One option would be to remove the CRT and yoke and put them in the front seat. But the whole thing might ride OK face down if everything's secure and you avoid potholes.
Phil Nelson
Phil's Old Radios
http://antiqueradio.org/index.html