Do you have a meter that can measure current? OK if it is DC current as you have a 3v DC battery.
I think it will be hard to find a cheap AC coil relay unless you get lucky at a swap meet or surplus store.
You are probably more likely to find a surplus or cheap 12v DC coil relay. In which case add a few 10 cent diodes (one unique part) and I think it would work. I can elaborate if diodes are permitted.