Try it, like a pants label. Are they te iron on ones with the round corners, or square ones? Round ones seem to stay on better. Would you consider writing your name across the nappy in laundry marker? Could be done on each of the layers if you got the snap ins.

Reminds me, we had an IttiLeak today, around the thighs. Annoying! Partly due to madam's epic sleep, but also poor fit, which is a bummer!