Yeah the metal lathe is what you want to drill the hole… you can pass the stock through the machine… what is the screw size? M5? M6? The gear cutting function of the lathe is broken unfortunately, but a tap will do just fine if we have the correct size…
I can help you tomorrow - but @Gergo_Dala is the expert! I am a newbie on the lathe so there is a slight risk I mess up your metal… 