Why is my flash underexposing only when using a wireless trigger?

It looks to me like the flash is pointed straight at the mirror when attached to your camera and may be pointed up at an angle when you are holding it in your hand.

Rather than trying to compare flash brightness by a reflection in a mirror that is very sensitive to the angle of the flash's beam, have you tried using it to illuminate a subject that does not include the flash itself?

Another thing you might want to look at is the focal length of the lens you are using versus the "Auto" setting on the flash (when attached directly to the hot shoe) versus "70mm" when controlled by the R2.

Is the lens you are using a 70mm lens? A 45-50mm lens? Something else? Flash zoom settings should be adjusted to "equivalent focal length" because the critical thing is actually angle of view, not focal length. Flash zoom setting affects the size of the cone of light projected by the lens, which in turn affects the density of light per unit area.

1/2000s of exposure time will give most remote flash triggers a run for their money. I recommend using a reasonably dark environment for testing (pretty much anything but sunshine should do) and putting exposure time significantly up, then dialing it down until stuff stops working as expected.

Does the flash trigger guarantee to work with HSS for off-camera flash? Some flash triggers stating to support various flash functions only do it for a flash immediately mounted on the trigger itself (basically, they do pass-through for the flash contact arrangement of the system they claim to support, but don't interpret the signals on their own).