If you can somehow get 4 kV out of a CCFL driver it should work. I don't think so!
A magnetron doesn't start conducting until about 3.9 kV, then it conducts heavily like a 4 kV Zener.
The safest way to do it would be to make a power supply by pulsing a car ignition coil, arranged so it puts out a positive voltage to the anode. You will need something like a scope soldering iron transformer for the filament, which takes 3.3 V & about 11 A.
If you want a smooth output, you will need a capacitor rated at more than 4 kV and a string of at least 5 1N4007 diodes as well. Preheat the cathode to limit the voltage before applying the HT.
Put the high voltage on the anode, as the soldering iron transformer insulation is not designed for 4 kV, or use the filament winding of a microwave transformer.
The other option is a 7.5 kV neon sign transformer and a bridge of 4 strings of at least 12 1N4007s (preferably 15, to handle the peak voltage if the cathode is not preheated). Neon sign transformers are rated at o/c voltage & s/c current, &, as far as I was able to find out, a 30 mA transformer produces about 26mA at a tube voltage of 1/2 the transformer o/c voltage, so it will give you about 24.4 mA RMS (about 22 mA average) at 4 kV. Magnetrons normally run at about 300 mA, so it will produce about 1/14 of the normal output power. Keep it away from your eyes though.
Check out this advice by Neil_UK on
Magnetron, and Microwave - Electrical Engineering Stack to someone else contemplating using a magnetron outside of the oven. It will not only give you a good laugh with his sense of humor, it is very good advice.