Copper wire will quickly deform even when not heated as you apply pressure. Further, as a very low resistance, it will not heat well given most power supplies, and particularly using batteries.
The internal resistance of the power supply will dominate the circuit, and most of the heating energy will be lost in the batteries or power supply, rather than the wire.
You might be tempted to choose a thinner wire to increase its resistance, but you will only be frustrated by the lack of mechanical strength in the copper wire.
You can still make this work if you support the wire. Adhering it to a thin support, such as glass, will allow you to heat the wire significantly without worrying too much about breaking it while using it.
Unfortunately you then run into another problem - oxidation and uneven heating leading to fusing. Even if you find wire that is perfectly conductive throughout and has no change in resistance along its length, over time it will oxidize unevenly, causing hotspots and cold spots, and in order to get the whole thing up to temperature you will likely find one spot get hot enough to turn into a fuse, and cause a break in the wire.
This isn't going to happen immediately though, so if copper is your only choice and you've solved the power supply internal resistance problem, then you can simply replace the wire whenever it stops cutting well.
However, there are many other metals with superior mechanical and electrical attributes that would better fit your needs.