From my copy of the 1904 H&H catalog, page 10, "Royal Hammerless Patent Single Trigger Gun", they clearly state that it enables the gun to be fired right-left or left-right, with the left barrel being fired first by pushing the trigger forward.
If you're not getting the right barrel to fire after firing the left first, I would suggest you need to get it serviced by someone qualified.
That trigger, like more than one English single trigger, was not exactly a roaring success. It has been a long time since it was used, no doubt for good reason.
Someone "qualified" to work on it might be harder to find than a gun so equipped.
A wonderful candidate for conversion to double triggers, in my humble opinion.
Best,
Ted