Prof. Barker,
As far as I know, nobody calls themselves Huhan Pai. It was a label that was invented at least by the early 1980s (I'm not sure if it was by Public Security, by Religious Affairs offices, or by local citizens and Christians) and used during the 1983 crackdown on the Local Church and other Christian groups in Zhejiang, Fujian, Henan, and elsewhere. Information about this crackdown is often included in the newer county gazetteers on public security (xianzhi gong'anzhi), now that (since 1999-2000) they sometimes include a historical overview of "anti-cult activities."
The label Huhan Pai was also included on a list of banned (mostly Christian) religious groups that was circulated in the wake of the banning of Falun Gong in 1999 (and later leaked on the internet). It continues to occasionally be used as a label for certain Chinese Christian groups that are viewed as being heretical or new religious movements, particularly those related to the Local Church and its offshoots, but it isn't actually the name of a particular group and doesn't have a lot of analytic value as a category, in my opinion.
Anybody you ask about Huhan Pai is likely to reject the label, though, because (1) it's a made-up label, and (2) it refers to a group that's explicitly banned, so no one wants to be associated with that name! That's also likely true for the names/labels of other explicitly banned Christian groups (Three Ranks of Servants, etc.).
That said, I'm not really an expert on this and my understanding may be somewhat out of date.
Best,
Jonathan Walton
PhD student, UCSD