Thanks for the replies. I'm thinking there is a way to have a permanent nichrome hot wire with a small battery, but it would take someone with more knowledge than me to figure out the details.
Wallace, what battery are you using? Perhaps your nichrome is not breaking but my nichrome is burning through because of a difference in the battery, since everything else is the same.
I've used several different LiPos but most have a C rating around 30. Keeping the elastic tight seems to be critical and making up a dozen or so, once you find the sweet spot, to keep on hand just makes things easier in the field. Personally, i enjoy creating/tinkering with things such as this much more satisfying than buying something.
There are some calculators that can tell you what gauge and length of nichrome is needed for a particular job, but to use them we'd need to know the desired temperature of the wire -- hot enough to cut the cord quickly but not so hot that the nichrome breaks, see
https://www.easycalculation.com/engineering/electrical/nichrome-wire-calculator.php and
https://www.jacobs-online.biz/nichrome/NichromeCalc.html (600F is recommended for styrofoam hot wire cutters).
@Edison: "It might be possible to reinforce the strength of the nichrome by wrapping a length of it around something." It is definitely the pressure of the cord on the nichrome that is causing my nichrome to break. I've fired it a few times with no cord and the nichrome turns red hot but does not break. Of course, we need substantial pressure to ensure the cord breaks, but Edison's idea to wrap the nichrome around a ceramic support could solve the problem.
@Charles: "I took a look, and I'm using 32 AWG. 10 ohms/foot and I'm using about an inch. Mostly with a Quark, but I have flown it with the Quantum, too." Good to know that 32 gauge works, it is substantially more durable than my 36.
@Tobor: "You can always use a load resistor, something in the 1W~10W and 1ohm~50ohm range." I don't think the nichrome will get hot if a resistor is in series.
@rocket troy and Charles: "there are a few avenues to possibly explore like using a small LIPO battery with a very high "C" rating for your power source thereby allowing the use of a thicker resistance wire." Agree that the C rating is critical, wish I knew what the rating is on my battery.
One guy made a durable electric cigarette lighter with 27 gauge nichrome and a pair of 3.7v Li-Ion batteries, although his batteries are quite large, see
https://www.instructables.com/id/DIY-Homemade-Electric-Lighter/