This is hardly a task for the typical Electronics Point member. Why do you want to design and build a do-it-yourself (DIY) power supply when there are commercial off-the-shelf (COTS) units available that are designed for the PE300BFA lamp? See your attachment.
I once built a power supply for a short-arc xenon lamp while working as an electronics technician for the University of Dayton Research Institute, shortly after being hired there in June 1967. By then I knew a little bit about power supply design, and almost nothing about high-pressure arc lamps. Xenon lamps were popular back then as illumination sources for movie theater projectors projecting large-screen 35mm color film images. The xenon lamp replaced the finicky carbon-arc lamp in commercial movie projectors, allowing for just one operator to service several projectors, mainly by changing film reels as needed. The xenon projector lamps were quite large and dangerous, but the risk of accidental breakage during lamp replacement was considered acceptable to theater management.
Most modern lamps DO NOT have external ignition capability by application of an external high-voltage pulse to a third lamp electrode. Instead, ignition is applied in series with the lamp by means of a low-resistance pulse transformer excited by a separate ignition circuit to create kilovolts of ignition excitation.
The reason for the low-resistance is the operating current for the lamp, which is generally several amperes, must also be carried by the pulse transformer secondary winding. You don't want to have a significant voltage drop in series with the lamp that is caused by high current through the secondary winding. So, finding a suitable pulse transformer and excitation circuit for same should be your first DIY priority.
Next, download
this document which describes how to use your PE300BFA xenon lamp. Pay particular attention to Figure 36 on Page 30, which is a schematic of a power supply (including ignition circuits) for this lamp. Good luck with that.