1.
A flyback operates best at a 50 percent duty cycle (or thereabouts). You can adjust this, but within limits.
2.
Even if the transformer primary has the same Henry value as the single coil, they need to be operated at different frequencies.
3.
While your project is in the development stage, try using a full-wave diode bridge at the secondary. This will take advantage of all current flows, in either direction.
Later you can substitute a single diode, when you see which orientation yields maximum current.