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.
James Clerk Maxwell popularized this method of dimensional analysis; in fact, before him, dimensions were a mishmash, unstandardized, and such analysis was impossible.
I'll give you silicon, which has specific-heat about 3X higher than tungsten.
At 2 picoJoules/(cubic_micron*degree Centigrade), suppose we want to short the output driver of a microcontroller? That driver is 100 micron * 100micron (its a powerful transistor), with 0.1 amp short-circuit ability. Assume 2.5 volts.
How long before the transistor reaches 1,025 degrees Centigrade? starts at 25C.
Our power is I*V = 0.1amp * 2.5 volts = 0.25 watts, or 250Billion picoJoules/second. The volume of silicon? Assume we'll only heat the top 100microns of the Integrated Circuit during our experiment (that depth has a thermal TimeConstant of 114 microseconds, and in that time "most" of the heat remains in that 100micron thickness. Our total volume is 100*100*100U or 10^6 cubic microns.
What is our rate-of-change-of-temperature? we want degrees/second as dimensions for our answer.
The only bit of info we have with seconds is the power: 4 seconds/joule
We want to cancel the "joules" so multiply 4seconds/joule by specific-heat of silicon
$$4 seconds/joule * 2 picoJoule/(cubicmicron * degree Cent) $$
Our rate of temperature rise is $$8 picoseconds/(cubicmicron *degree Cent)$$
And we have 1Million cubicmicrons of silicon. We need to cancel 'cubicmicron' in our answer, so multiply the answer by 10^6 cubicmicron, and we get
$$8 Million picoseconds/degree Cent$$ or $$8microseconds/degree Centigrade$$
We wanted 1,000degree Cent increase in temperature, thus 8,000 microseconds or 8 milliSeconds is the answer.
We initially assumed ALL THE HEAT would remain inside 100*100*100 micron cube.
In 8 milliSeconds, heat will have moved outside the cube.
A different method is needed for a correct answer.
And thank Maxwell, also the investigator of viscosity, for this method.
Best Answer
I don't believe you can do this with any degree of precision : it depends too much on the wire and its environment which determines its rate of cooling.
This 450 page book would give you a brief introduction to the difficulty; the TL/DR version is: Don't.
Which leaves you with the alternative problems of: