TBILISI, Georgia—Aleko Grigolashvili was just 11 years old when he and a friend ran away from home to join Georgian fighters battling Moscow-backed rebels in Abkhazia, a separatist enclave on the Black Sea.

The adventure barely lasted a month in the early 1990s, before Aleko returned to his panicking grandparents, and to school. But it set the tone for a lifetime of war against Russia. In 2008, Mr. Grigolashvili, by then an officer in the U.S.-trained Georgian military, fought Kremlin-backed separatists and regular Russian...