> To this day I don't know what the word "caucus" would even correspond to in Russian.
It would vary depending on the meaning in English, too.
The kind of caucus that nominates candidates would be "выборный съезд".
Congressional caucus is actually trickier, just because there's usually no close equivalent in other parliamentary systems (including the Russian one). It's like a political faction, but 1) its platform is not all-encompassing, and 2) its membership is not exclusive (i.e. people can, and normally do, belong to several different caucuses). For that, I don't think there's any good word other than loaning the English word directly.