bean I did read the whole of your OP, and I can see/hear what you are saying, but you are not really listening to what I am saying. Stop trying to "make" her do things, as Starlight says, let her lead you, which by the sound of things is essentially what you have been doing with most things except sleep, which you have been trying to take control of. A tired baby will fall asleep, where ever it is, in a pram, in a sling, on your lap in a restaurant, in a highchair, I agree with Starlight that "over tired" is an overused term, they can be over stimulated, but if you settle down some where a quiet and nurse her she will almost certainly fall asleep if she's tired.
And I think that babies who "self-settle" only do it because they have been taught to, and this invariably involves being left to cry, being given a dummy or put in a rocking/vibrating swing/chair. Small babies instinctively need physical contact. When tiny babies are put down and left alone it distresses them, some cry, some shutdown and go to sleep. CC works on exactly this loss of hope/shutdown mechanism. They don't stop crying and go to sleep because they are content and "settled" but because they see no alternative. In this respect "self settling" is a coping method, not a sign of contentment.
My messages may not seem that supportive, but as someone with two "non-sleepers" I know just how debilitating exhaustion really is, and how frustrating it is when you hear about 12 hour/7-7 sleeping babies, and how tempting it sometimes seems to go down the CC route. But if hearing her cry leaves you howling in the kitchen then I think you have answered your own question about CC being for you. I also know how much pressure you may well be under from well-meaning family and friends who think you need to get tough and deal with her "sleep problems", I know, I really do!! (my sister said to me "you'll have to leave her to cry eventually, you might as well get it over with" she was wrong, I have never left either DD to cry, and both now sleep all night in their own beds, it has just taken longer for them to do it than most people consider to be acceptable/normal)
I used to sit with them on my lap until they were done cluster feeding then, once they were deeply asleep I'd transfer them to the reclining chair thingy, or DP would hold her. We found it really nice, to be cuddling a sleeping baby, and not intrusive at all. If you really can't do this then take one side off her cot and wedge/cable tie it to your bed and lay her down in it and lie down next to her and feed her to sleep. Once she's asleep you can get up with out disturbing her the way being put down does, and she'll probably stay asleep. This sidecar cot worked really well for us with DD2 because she is quite a light sleeper, and if she was not moved after she fell asleep she would stay asleep, but putting her down when she fell asleep on y lap was really hard, she always woke. (you could put DD1 down once she was deep asleep, she sleeps like her dad, out cold!!)