Thanks for the file, P.
From my point of view, the geometry of the flower is not ideal to just add a Cloth or a Softbody to it. It exploded in every single case I tried.
The following exploration was to build a cage around and use dynamics on that. Also, here, the results are not usable, from my point of view. (I had built the cage via Volume, Remesh 10%, then used in a Mesh Deformer.)
As I have said often, Dynamics are not the tool for art-directing delicate shapes or situations. The amount of time one has to invest in fine-tuning the results is often just not worth it.
I used the Melt Deformer and others to see how this one might support any efforts. I don’t think that doesn’t go anywhere.
Since you already have a joint rig established, why not use it? To create the final scene as well with it. Then use a pose morph to “model” the changes? If that was from the start of the plan, I might have tripled each joint (three instead of one each time) to express that appropriate to have the “final” moment” of the flower.
Or much more straightforward, see the file, use an FFD [deformer], and animate that cage via a PoseMorph (use one slider, then add the next with the selection “In Place”, so each slider shows a step along the way. My example can’t be as detailed as you would for your production; for me, it is more to show the how. I don’t have the script here, speaking of the artist’s intention. You will know how it has to look.
Example:
https://www.amazon.com/clouddrive/share/k2XgE9XBO5qI7RfQJJZEGaQQVpYNJgBzPmsMn7z96k
There are other ways, but since you have a joint rig, I think any point manipulating options/functions would only add more complexity to it than be of any actual use.
All the best