I looked up in Wikipedia and found a different story. The Iroquois nation cleaved during the Revolutionary War, one part siding with the Brits, the other with the Americans. Other tribes did the same. When I scanned, I couldn’t find an example of a faction who unwisely chose to fight both sides.
War via Proxy
Modestly, I think my cold war example is the best. The US, China and the Soviet Union all fought each other in 1969. The problem here is that the Soviet’s support for Vietnam was mostly material: fatalities among their advisers were …16. China suffered 1446 dead, 4200 wounded; the US had 58,220 dead and 303,635 wounded. During the Sino-Soviet border conflict, 58 Soviets were killed and 100-600 Chinese died.
Dates: The Chinese-Soviet border conflict occurred on March 2, 15, 17 and 21 1969. Nixon had taken office that January. Nixon’s secret bombing of Cambodia began on March 18, 1969. Surprisingly, it was noticed.