I'm also an avid hammock sleeper, and I've given it considerable thought. My conclusion is that it's a real bad idea.
First off, it puts a side load on the tie down ring which wasn't engineered for that.
Second, the vector force of a load suspended between two distant points is considerably higher than the load itself. Every time you jostle and bounce while getting arranged in the hammock you significantly increase that vector force. Your 200lb body could easily exert 600lbs of force on your tie-down ring, in a direction it was never meant to be stressed.
Third...you're going to have to be tucked up real close to a tree, or string out a lot of line and then pull it all taut enough that your but doesn't hit the ground. Being able to park that close to trees isn't real common, especially for a airplane that's too heavy to really push around by hand. And the more distance between anchors and the tauter you pull the line, the higher the vector force being applied to your...er...vintage airframe. You could top a thousand pounds of load on your tie down ring real easy.
All that said, it might be fine. I just don't see any real upside to finding out, and a lot of potential downsides. My hammock has an attached rainfly, so I don't see any advantage to being under the wing, and honestly, condensation or rain running off the wing often makes it a worse shelter than just being out between a couple trees.
Just my take...14 years of hammock camping out of an airplane.