Problems driving pulse transformer with low-side MOSFET switch
I wouldn't use a flyback converter for that application.  You would probably do better with a forward converter with a reset winding.

In general you need feedback from the output to do a flyback as the duty cycle required depends upon the load.  Since you have an isolated output it is difficult to get that feedback.

Here is a paper describing such an implementation.Single Switch forward converter.

Notice that a second winding is used with a diode to return the magnetic flux to zero after each cycle (often referred to as flux reset).

You may also be able to use a flyback converter with feedback coming from a third winding (the transformer you selected has 5 windings).

On your schematic you don't show any capacitor or load after the diode on the secondary.  Also you don't need a bridge rectifier - just a single diode is needed.

Here is an example using an optocoupler-coupler