I think the inverter transformer is meant to be used with an actual inverter. By itself, it will not produce any AC.
Also, you need a rectifier before C1; otherwise C1 will short your transformer secondary.
Finally, you need some way to trigger the pulse transformer. Simply adding a capacitor across the primary of T2 will not help. You actually have to add a circuit which will periodically discharge C2 into T2 primary to get the high voltage to trigger the Xenon tube.
Jem