Don't choose my answer. Wait for more knowledgeable people. I just want to give some suggestions.
First, building muscle is very difficult. You shouldn't fear becoming bulky, it will not happen unless you do a very specific training and nutrition during a long time. Just exercise and forget about that fear (your muscles may become denser and harder in the long run, which is very nice, but not bulky).
Second, if you have loose joints, I don't think it is a good idea that you emphasize passive stretches. I think it is the last thing you want to do.
Third, I think it is a good idea that you focus on slow exercises with perfect form. That helps having total control over your movements and so you can avoid things that go wrongly out of place. It might be lifting weights or callisthenics, or Tai chi or whatever you choose, but always it would be something made up from slow movements where you can retain full concious control.
If you lift weights, choosing exercises involving many stabilization muscles around your joints is much better than doing isolation. With loose joints, you really need to strengthen your muscles around the joints because they must provide extra stability. For instance, people with ACL ligament injuries rely more on their quadriceps and hamstring muscles to provide knee stability. They prefer squats and single-leg deadlifts to other options. If you look at the model in the last link, you will see that she has to activate a lot of muscles around her knee in order to maintain equilibrium. That is the idea. Dumbbells better that barbells or machines, standing better than sitting, and so on.
Yoga is also a good option. It is slow enough so that you retain full control over your movements, and it also makes you quite concious about your joints. Tai Chi is also a good idea. Engage in a good Tai Chi class and forget about the rest. You can try to switch from Tai Chi to Kung-fu later, they are closely related. Designing your own training can be very frustrating if you are prone to injuries.
I hope this helps. In any case, I think the slow, controlled execution is the most important tip.