The front light adapter plate position can be adjusted what also changes the position of the stopper plug, maybe changing the position of the plate would help ?

On the Birdy without front light, the plug is different and can be adjusted (at least the one I have mounted on my Ti Birdy, but this bike is different from a Birdy III, so I am never sure that it will be the same on a Birdy III).

Now what you could do to mount the stopper plug on the right side in the fork threaded hole is to put in the thread of the stopper plug a short threaded rod with one or two nuts and use the nut/nuts to adjust the length of the stopper lug.