In “The Man Who Would Be King,” Danny and Peachy are both Masons, in addition to the narrator. In "Kafiristan," what role does this membership play in their adventure?

The fact that Peachey and Danny are both Masons is what ultimately enables them to command power over Kafiristan after a somewhat rocky start. Some of the villagers have already decided that the two men should rule them—"they think we're gods"—but it is the unexpected discovery of Masonry among the locals which enables them to ascend to true power when Dravot declares, "I don't want to fight no more."
As it is explained to the narrator, Dravot brings forth an old Chief they called "Billy Fish" and suggests that Peachey shake hands with him. They are astonished to find that the chief gives them "the Grip," a Mason's handshake. Trying the "Fellow Craft Grip," this too is answered properly, and the men are baffled:

"A Fellow Craft he is!" I says to Dan. "Does he know the word?"
"He does," says Dan, "and all the priests know. It’s a miracle! The Chiefs and the priest can work a Fellow Craft Lodge in a way that’s very like ours, and they’ve cut the marks on the rocks, but they don’t know the Third Degree, and they’ve come to find out. It’s Gord’s Truth. I’ve known these long years that the Afghans knew up to the Fellow Craft Degree, but this is a miracle. A god and a Grand-Master of the Craft am I, and a Lodge in the Third Degree I will open, and we’ll raise the head priests and the Chiefs of the villages."

While they are at first slightly concerned about the legality of running a Masonic Lodge without the Lodge having been properly commissioned, Dravot insists that they shouldn't ask too many questions of the miracle, which "means running the country as easy as a four-wheeled bogy on a down grade."
So, the men go about making the Lodge "as regular as possible," showing the priests' families how to make aprons of the Masonic degrees and setting up the lodge with stones. As well as their adherence to Masonic ritual, however, luck plays a significant part in the men's success:

. . . ten priests took and tilted over the Grand-Master’s chair—which was to say the stone of Imbra. The priest begins rubbing the bottom end of it to clear away the black dirt, and presently he shows all the other priests the Master’s Mark, same as was on Dravot’s apron, cut into the stone. Not even the priests of the temple of Imbra knew it was there. The old chap falls flat on his face at Dravot’s feet and kisses ’em.
"Luck again," says Dravot, across the Lodge to me, "they say it’s the missing Mark that no one could understand the why of. We’re more than safe now."

After this incident, they declare themselves kings of Kafiristan, and begin to "raise" those from the outlying villagers who were considered "worthy" into Masonry. Many of those who are taught the first degrees seem to know them already; meanwhile, spreading Masonry throughout the villages serves to minimize their likelihood of infighting. The system means that the two "kings" are, at this point, held in utmost respect as the guardians of the system.