In reply to bestofbritishjanet:
Hey, I'm a hypermobile climber too and have had exactly the same thing as you have described. This reply might turn into an essay, but to try and explain the experience I have had, I'd say it seemed to stem from a number of areas.
1. Arm strength
2. Shoulder Integrity
3. Back (Rhomoid) strength
4. Pectoral Strength
You said you have a strong chest, so that pretty much rules out number 4, and I would also say, from personal experience, that arm strength is not actually that big an issue. I had to concentrate on shoulder integrity and my back muscle.
Basically, whenever I do chin ups or press ups or anything that involves a significant portion of my weight being in arms/shoulders my joints stretch and lose their integrity. What I had to work on was the strength of the muscles which allow me to hold my shoulder blades down and in as this has the knock on effect of keeping the head of my shoulder back and in line with the rest of my body and makes it harder for my joint to overstretch as actually a lot of the weight is distributed across my back, rather than in joints which will sublux if I pick up a shopping bag too quickly.
What I did to gain this strength was not very different from normal climbing training, but I suppose I concentrated on different areas and techniques while doing so.
Traversing (especially on features walls and non juggy holds) has really helped with my hand and back strength. Any types of traverse route which require you to use some kind of compression, or finely balanced weight moves seems to have the added bonus for me that it keeps my shoulders down, with less chance of overloading a joint, but while being able to keep sustained tension in the muscles through my back. Has helped immensely.
Dead hangs – making sure that you're not using tiny edges and running the risk of dislocating fingers – are good for arm strength and stamina, but I also do reps of a certain number of seconds dead hang, and then finishing with slow controlled 'rolls' where I concentrate of moving my joints back into place to the point where I can engage the muscle and bend my elbow again. After a few months of doing this, I am now at the point where I can comfortably dead hang on slightly bent arms for good times, without my rhomboids/deltoids screaming after a few seconds. Seeing that when I do this my shoulders are not stretched was a lightbulb moment.
Planking. Even though most people use this for core, the length of time that you are sat on your elbows will, for a lot of hypermobile people anyway, feel like someone is sat on your neck as all your joints shift out of place. Doing shorter but more repeated reps of planking, all the while concentrating on holding my shoulders as square as possible, while keeping more core firm, has helped a lot.
Anyway, there is loads more stuff that I have found, and I could go on but I won't. But please feel free to get in touch if there's any other way that I could help.