MOSCOW—When a Russian biker gang known as the Night Wolves wanted to settle a score, they didn’t reach for knives and knuckle-dusters. Instead, they took on their rivals in the traditional Russian fashion and denounced them as foreign agents.

That is what happened when the Night Wolves, the country’s most prominent motorcycle club, got into a feud with local chapters of the Hell’s Angels and the Bandidos. Last year, the leader of the Night Wolves, Alexander Zaldostanov, sent an angrily worded letter to a lawmaker, asking for...