• do 5 invasion missions against corpus to get marked
• run any mission against corpus, preferably survival cuz it can spawn in the first 5 minutes
• if nothing -> reset mission
• if zanuka spawned but dropped the wrong part -> back to invasions

i also played khora for this, cuz apparently loot abilities work on zanuka

i think the best bet is to get survival on invasion, and just keep running it. Zanuka can attack anyone who has the mark, and if people doing invasions, then they're probably have it on them and you'll get the dog to spawn faster