the percentage change in resistance is what determines the relative change in signal level
nichrome (as was originally suggested) which has 18 times the resistance of tungsten. But with only 0.0004% change per K it would be hard to detect a 5 degree change. It's the percent change that determines how easy it is to measure a small change in temperature.
The low resistance of copper just means you will need to use a very small copper wire (such as a single strand of a 28AWG stranded copper wire) and wind it into a coil to get a reasonable resistance to measure with.
That is not to say that tungsten won't work. Since he only needs +/- 5 degrees accuracy, that would cause a 0.4% change in the tungsten resistance. You should be able to reliably detect that with reasonably careful circuit design.