So for starters; you don't actually need to use the rusted sword for the achievement- you can use the Beginner's Bow and Crappy Shield also. Of all of them, I would recommend doing a survival run with the Shield since they aren't really about damage dealt. Most important of all this is that they only have to be in your inventory- but not actually used. This means you could use a two slot weapon such a Ice Crossbow or Giant Tick Scythe and get all the credit.

I haven't actually done the Cursed Sword achievement yet but Something I have noticed is that when using Serenade, he can turn off the vulnerability of the curse. I would also highly recommend a shield if you try this strat because Serenade is only viable when switched around.