Cherv.... you do not use two regulators. In your case in order to use the weber grill Q120 you will need to place a by pass valve before the TT regulator. It will require some parts rearrangement in your current TT propane manifold. I did this to mine too....so I can have both a regulate exterior connection and non regulated connection.
Here is another thought you might not have pondered...you can refill those little bottles yourself. YHEP....you sure can. I have this for my Coleman stove and it works well.... here is a video.