Flyback Converter - Low Voltage High Current Output Problem
You are on the right track with your edit: you will have to decrease the number of turns on the secondary to run at higher currents, or get a core with larger cross section.

A flyback converter can be viewed as two inductors sharing the same core.  You store energy in the inductor core by energizing the primary winding, then dump it into the load through the secondary winding.

If you are trying to get 600 mA out of the secondary, the average current in the secondary winding has to be 600 mA.  If you are at, say, 50% duty cycle, the average current during the output pulse must be 1.2 amperes.  And, since the output current pulse waveform decays via a ramp down to zero amperes during the output pulse, the peak current  on the output must be around 2.4 amps.  The energy that is stored in each pulse is proportional to the square of the peak current for an inductor.

Now see what happens if you start to increase the flux density in the transformer.  The transformer becomes less efficient, and the output suffers.  The flyback loop reacts by increasing the duty cycle of the FET "on" time, which reduces the the duty cycle of the "off" time, so the peak output current must increase further.

I would try to remove turns from your secondary winding until it just barely works with your highest-voltage load, then see how much current you can get out at lower voltages.