Hot wire cutter. Can I use copper wire
Not really, it does not self heat very well and when hot it anneals and gets soft.

Nichrome is good, steel or stainless steel should also work ok and may be preferred if you need less heat and more strength. Sources of resisting wire would include fishing leaders (or other fine wire rope), piano wire, guitar strings (music wire), salvaged heating element wires (brittle if previously heated), florists wire, and pretty much anything except copper and aluminium.

Using 3 AA cells to heat wire will be disappointing.  They made some gimmick bag sealers that heated a short 10mm length of nichrome wire with 2 AA cells but I doubt it would have got hot enough to cut Styrofoam cleanly.

If you should find some very fine filament that will get hot enough to melt Styrofoam with the 3 cells you may find that it will cool down in air and on contact that your cutting speed will be measured in frustration.  Transformers supplying 6 to 24V are often suggested for the wire heating and this for lengths of about 300mm (12 inches). The current may be a few amps so would drain small cells fast. Note that you need an isolating transformer for your scenario - using an autotransformer may have dire consequences (including electrocution).