Zhampir is right, there are examples aplenty when there were two organisations doing exactly the same thing. For instance, in 19th century Russia there were effectively two police agencies with many of the functions crossing over. Same goes for KGB and GRU, with the divide coming in the form of the first being civilan, and the second being military. Then take Italy, for instance- there is the Polizia, and then there is the Gendarmeria, which practically does the same thing. The argument for having two organisations is that they'll try to outcompete each other, but this strategy oftern backfires when they simply start suspecting each other and preventing the other organisation from carrying out their plans.
Nevertheless, I think that there is a possibility for Spriggats getting accepted into the Cult as time passes by, I am just unsure wherether a few decades is enough after millenia of division (and taking into consideration that these beings live much longer- a few hundred years, by the looks of it, so generation changes and historical events fall into oblivion much more slowly).