Surely Abkhazia could be considered for "Russian republic in the North Caucasus"? I'm not particularly knowledgeable about the topic, but Russia was the first to recognize Abkhazia officially and has occupied the region despite it being in Georgia de-jure. Also, in a more literal sense, they are a republic ("Republic of Abkhazia") in the North Caucasus that speaks Russian.
I don't think I will add it now. Russia recognizes Abkhazia as a sovereign state, not as part of Russia. The population is majority ethnic Abkhaz, and though the Russian language is official it seems Abkhaz is the primary language. So I don't think it would be correct to call it a "Russian" republic