Tai Fu my friend great outside the box... thinking from you
I think that one of the possible reason why folks don't have the router base riding on the guitar side is that it would also need to register on the top and back. It's possible that anything registering on the soft guitar top, think Western Red Cedar, would crease it leaving a line.
You can make a donut which is a wedge of wood, I used MDF for mine, that is calculate to be the angle required to permit the router to sit up straight when sitting on the dome of the back and top. Of course both the top and back would be different angles unless you built them to the same dome. This donut attaches with double stick tape to the router base and then has a mark, visible, made at it's thinnest point. As you route around the guitar keeping the mark that you made always pointed toward the center of the guitar the donut negates the dome's ability to make the router tilt backwards. This makes for a more uniform cut all the way around the guitar.
Here are some pics of the donut that I made and used for a while. It worked very well.
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