You should use ceramic insulators to separate the hot cutting wire from the "Y" handle, allowing the cutting wire to reach a higher temperature. It is important to use a cutting wire of significantly higher resistance than the copper wire providing its current. You should terminate the cutting wire in a highly conductive clamp (insulated from the "Y" handle if the handle is made of metal) to which you also clamp the copper wire that energizes the cutting wire. Aluminum is inexpensive, but a hard-copper terminal block would be better. You can drill holes in the block to accept the wires and use set-screws in threaded holes drilled perpendicular to the wire holes to secure the wires. Even better would be a split-clamp arrangement similar in design to an automobile battery clamp that tightens down on the wire(s) with a hardened machine screw or hex-head bolt. This requires cutting two slot on either side of the wire holes to allow the holes to close on the wires. The advice of a machinist would be helpful in the design of the clamps, but this isn't rocket science. The idea is to minimize resistance in the connections and the wires supplying power, while at the same time making possible easy removal and replacement of the wires. You want very low electrical resistance everywhere except in the cutting wire.
Cutting wires can be made of stainless steel or nichrome, both available in various wire gauges from supply houses such as McMaster-Carr. The wires will eventually break from the cyclic heating and cooling effects on their internal crystalline structure, so purchase the wire on spools and cut-to-length as needed.
The easiest way to apply an adjustable current is with a variable AC transformer or Variac connected to the primary of a largeish step-down transformer providing about 12 to 24 V AC on the secondary with a current rating of at least ten amperes. A higher current rating is better.The operating current will depend on the cutting wire composition, wire diameter, and wire length. Some experimentation is necessary to determine the voltage and current you need to heat the cutting wire sufficiently to go through Styrofoam easily. Make sure that you use oversized copper wire when determining the cutting wire voltage and current parameters. You can then use published tables to find out what wire gauge is appropriate for the "final product." Bigger is always better, but you need some flexibility too. Use welding wire, which has a much higher strand count for a given gauge than ordinary wire, lending it much better flexibility.
I think the key to this project is a good design for the handle that allows a semi-circular loop to be secured in the loop crotch without melting the handle or causing discomfort to the operator. You will probably need several handles of different sizes to accommodate different wire loop sizes. Be creative in your search for ceramic insulators. Many types are available as inexpensive replacement parts for use in electric clothes dryers.
