It would be intersting to know why you failed with *Cloth*. I would also try it with *Cloth* but with a *Mesh Deform* modifier:

- Create your ear/tail. Create a rather simple box that contains it, make it transparent in *Solid* mode and invisible in *Render* mode, parent it to the armature (here I parented the 2 objects to an empty):

[![enter image description here][1]][1]

- Weight paint the box, make it gradient red/blue from bottom to top:

[![enter image description here][2]][2]

- Give your ear/tail object a *Mesh Deform* modifier with the box as *Object*, click on the *Bind* button:

[![enter image description here][3]][3]

- Give your box a *Physics > Cloth*. In the *Shape* panel > *Pin Group*, select the group you've created with the weight painting:

[![enter image description here][4]][4]

- Move the armature, the box follows and bounce, it makes the ear/tail bounce as well. Change the *Cloth* settings to make the effect more elastic or stiff, etc.

  [1]: https://i.sstatic.net/299mv.jpg
  [2]: https://i.sstatic.net/mvF57.jpg
  [3]: https://i.sstatic.net/uJhC5.jpg
  [4]: https://i.sstatic.net/nvB1K.jpg