Roughly speaking, an electric heating element rises in temperature until it reaches a state where the power input (3kW in your case) is equal to the power output (heat into the water).
A heater designed for immersion in water is built so that water will keep it cool enough not to be damaged at its rated power. The water will usually be driven by convection to move constantly past the element, carrying away enough heat to prevent the element overheating. Once the water reaches boiling point, the temperature of the water stops rising, but the conversion of water into steam uses lots and lots of heat, which the element replaces. At this point the element will be significantly above the temperature of the water (it's this temperature difference which drives the heat out of the element into the water), but its temperature will be roughly constant.
Things will stay like this, until enough water boils away that the element starts to become exposed to air. The air is a much worse conductor of heat than the water was, so the element temperature needs to rise to re-establish the '3kW in, 3kW out' equilibrium.
An element designed solely for use in water will not be able to survive this new, much higher, temperature, and will be damaged very quickly.
So the short answer is, you can supply 3kW to a 3kW immersion heater for as long as you like as long as it is immersed and not a second longer.
The element on your electric stove is also designed to run continuously at its full rated power (at 'max', it will not be pulsed), but it does this by being longer than a typical immersion heater of the same power (so more surface area to lose heat from) and being built to run at a much higher temperature without failure.
A two thousand Watt transformer is going to be big and heavy. For AC conversion the transformer is a good way to go though. They are efficient and produce real AC waveform outputs.
It is possible to envision a design that would convert the voltage by rectifying the input AC to DC. That would then be converted to a different voltage using a high frequency switcher design and then synthesize a fake AC output. Such device would not be trivial to build however but it could be smaller due to the smaller magnetics required.
Personally I think you would be way better off to set your 220 volt heater aside and get a replacement product that is designed from the outset to work on 110V. And surprisingly that may even cost less.
Best Answer
I did something like this. The approach I took was a bit different. I created two components that had a single pad each. On the schematic, I connected these components together with a trace. In the layout, I created a serpentine trace to join the two components together. DRC was happy. Everything was fine. Instead of having a heater component, I had a heater PCB.