Bob. Or maybe Robert if you're being formal. Unless of course their name is really Margaret and for some reason people call them Peggy, which is something I've never understood. I mean, seriously? It's like calling Richard "Dick". The only commonality is a random vowel.
Anyway, I digress. Warforged works because it's simple and evocative, unique without being totally made up. They were forged for war, it's right there in the name. So maybe you need to start with their origin story. If they were not created to be warriors, why or how were they created? Are they still being built, and if so why?
Barring some bolt from the blue, you could just use a standard term like androids, automatons, mech or from sci-fi synthetic humans aka synths. Robot for example comes from Slavic robota for "forced labor".
So in my world I might use google to look up "slave labor" in latin to give me "servi laborant" which could become Serborants or Serlab? I don't know, neither really rolls off the tongue but you get the idea. An option is to just make up a new word that you think sounds cool, it doesn't need to mean anything. Maybe they're just named after their inventor? Kind of like how we call Frankenstein's monster Frankenstein?
Other options would be more how they were made. Soulforged, Constructed, Metal Men, or just Forged Ones.
But seriously? Jack for John? Really?