How greatly can heat increase resistance

The internal resistance of a D-cell is about 300 milliohms according to Energizer. That, with the 3.21 ohm wire resistance, should be generating about 9/3.5 = 2.6 Watt (P=VI = VV/R). I'm assuming you are using the batteries in series (to get 3V).
That won't be enough for the temperature rise in the box you are looking for although the wire should get warm/hot to the touch.
To get an idea about the amount of power required, have a look here: https://what-if.xkcd.com/35/

Some more details...
The thermal conductivity of cardboard is about 0.21 W/m/K. So it will conduct 0.21 W for every square meter surface area for every degree Kelvin in temperature difference for every meter thickness. To get the inside to a certain temperature, you have to pus as much power in as the box conducts to the outside (I'm ignoring radiation here as the temperature is relatively low).
Let's assume the area of your box is 10x10x20cm, which is 0.002 m2. Assume also it is one millimeter thick.
That means that to obtain a 15 degree Celsius temperature difference with the outside, you have to put in 0.21 * 15 * 0.002 / 0.001 = 0.21 * 30 = 6 W of energy.
As such, you need about 2 times more power and you should halve the resistance (length) of the wire.

The temperature control is a well-trodden path. You need to have a sensor of adequate accuracy and a heater of appropriate power level and a temperature controller. I would suggest purchasing a Pt100 sensor (100 ohm platinum RTD), a cartridge heater and a commercial PID temperature controller.

Have a block of aluminum fabricated with cross holes for the sensor and the heater. Bury the sensor inside the block so the tip is many hole diameters in. You could use copper too, but the machinist will be less enthusiastic about the deep holes in copper. The idea is to make the plate thick enough and thermally conductive enough that it is effectively isothermal to the degree that you care.

The heater does not have to be very powerful for this application, maybe 100W.

For the best accuracy you will want to keep air currents off the surface of the plate, so a cover would be a good idea. Insulate the bottom of the plate and use stand-offs so that even if the heater stays on at 100% it cannot cause a fire.

This should cost maybe a few hundred dollars, more or less, depending on quality of control, sensor accuracy etc. and you should be able to hold the temperature steady to within a couple tenths of a Kelvin.