Unfortunately, adding ambient lighting negates the whole point of using HDRI. I'm not sure, but I think the deal with the reflective objects working inside a transparent object is the reflective object isn't actually recieveing any light from the HDRI, it's just reflecting the HDRI. If you put 2 identical objects in a scene, shade one chrome and one plastic, set HDRI as your only light source, and turn Sky Light off in render options, the chrome object will look look bright and shiny (odd), and the plastic one will be left in the dark (as expected). Turn Sky Light on, the chrome object looks just like it did in the previous render, and now the plastic one will be illuminated (as expected). Again, I could be totally off base here. If any of you guys that understand the underlying physics of this stuff have any ideas as to what might be going on, would appreciate the input. P.S. I also tried turning off "cast shadows" on the tranparent object, made no difference in the final renders.