Hmmm.... the core does not dictate the power really.
What it does give you is a winding window. The core will give you the turns per volt at a particular frequency and magnetic strength... thats it in a nutshell.
After that it is up to you and the physics to get the wire in there that will dictate the copper losses... this governs the power you can get in a time period... ie massive power for very short periods... and lower power for extended periods..... and every where in between.
We choose a bigger core for higher power simply to get thicker wire in there. ..... ie the more core...... then the less turns per volt we get, and so the thicker wire and less turns we can get away with in the winding window.
So I will assume that all the losses will be in the copper ( not entirely true... but...), and we are looking at the I^2 R for the watts lost in the winding as heat. This is what will burn it up or not.
If we can get the heat to dissipate faster ( fan cooling, oil and fins etc etc), then we can loose more voltage due to current and the copper resistance, and still keep the temp in the safe range..... ie our watt rating will be higher with better cooling, or thicker shorter windings or both.
The iron loss will be the same regardless of power.... ie once we have wound the thing, and it does not saturate, then higher currents or loading ( even overload ) will not saturate the core... only more voltage or lower hertz will do that... not current... so it is mainly copper losses we have to worry about. Torroid steel is usually very good.
So answer to your question as I see it is how low can you get the copper resistance, verses how fast you can get rid of the heat due to copper losses... otherwise known as how long is a piece of string really.... gut feeling is bigger is better
I guess if you calculate the resistance of the primary you envisage and the secondary you can get in there, then you will have a good idea of how much power will be converted to heat...... that you have to get rid of at different currents.
It all comes down to efficiency, and getting rid of the heat.
Re reading that, I'm not sure I have helped all that much
but the long and the short of it is.......more copper is more goodderer
..........oztules