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PITFALL! https://youtu.be/SoZV8D13WuY?t=4m18s 
Rofl, just kidding, I very much know the feel.
So, I’ve read this thread and was curious as well about this whole rope swing thing. I loved some Spider-Man’s games.
So then I took a look around the internetz and came across this:
(lemme me pretend I understand whatever this dude says)
https://.com/watch?v=8vXDZgHvZco
Judging by the videos you’ve posted you are using physics constraints, attaching the character to the physic handle; So, I came up with a test based on the same setup…
On the PhysX simulation, the physical body will swing based on it’s velocity around the constraint so most of the pendulum math is eliminated by that already which is awesome and makes things much easier.
However, as time pass by the body will always lose velocity and stop moving no matter what, as it’s supposed to be. But for characters that is not ideal since you need a way to make the character ‘swing’ using his body’s weight; just like Spidey does.
What is missing on you character is the pendulum movement influenced by the way a person would manipulate it’s center of mass to gain velocity instead of slowly stagnating at the bottom of the rope system.
Notice that the closer the ball is to the Z axis, the more velocity it gains and that energy keeps the pendulum going. So in my test I’ve added that into consideration to ‘generate’ the pendulum movement.
You could try something like this and see if works for you; whatever the velocity the character is, the pendulum should take it and add to that direction only where gravity creates tension between the handle and the character’s body (when character is in certain angle to the handle and always below it). Hope it helps, good luck 
https://.com/watch?v=jnf7an5BOuk
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Wow, This post is a treasure chest of goodies! Thank-you so much! Let me mess around a bit and report back!
Seriously, I never thought I’d come across something that would not only blow my mind once, but multiple times in a single post! Thanks :>