I have no solution for a glass material. Changing normals of the inside wall of glass or using a different material for the inside of the glass results in problems with refraction etc.
Instead you might try to change the material of the object inside the glass bubble. I've put an Emssion Shader on Suzanne so that the reflection is clearly visible. If I mix the Emission (or any other) Shader with a completely white Transparent BSDF and use Glossy Depth from the Light Path input as a mix factor, then the reflection only shows the transparent material i.e. nothing.
Of course that's not very handy if you try to put objects with many different materials into the bubble because you cannot simply have one glass material with the correct settings, but many other materials to prepare.