Thank you for that. Now we know we are not having a "conversation" with a rank amateur.
I, too, am an electrical engineer with experience in embedded microprocessor control and data acquisition... as well as power electronics and driving "photoflash" devices... among many other things. I don't profess to be an "expert" in anything, but all my life have sought to learn new things and teach others who are interested what I may have to offer. If that is a character flaw, I will continue to live with it.
I don't understand why you are adamant about sine-wave drive in the secondary. The secondary waveform is passed into a voltage tripling rectifier circuit and the DC energy stored on capacitors. The rectifiers and capacitors don't much care what the driving waveform is. On the other hand, perhaps your concern is harmonic interference if a sinusoidal waveform is not used?
Here is a web page that describes the 1954-patented Bright and Royer relaxation oscillator, modified further down the page for sinusoidal oscillation by adding a series LC circuit and substituting MOSFET power transistors, which of course did not exist in 1954. If you insist on sinusoidal waveforms, a toroid transformer could be an efficient choice if its core characteristics support the frequency of oscillation and it is large enough to support the power transferred. Note that very little power is consumed after the capacitors are fully charged, until they are discharged through the flash tube.
Your original sketch showed a pulse transformer (presumably) but provided no information on its characteristics. Is this something you have on hand that you hope will be suitable? Is there a part number or other identifying information available? Pulse transformers are not generally used for sinusoidal waveforms.
You stated your circuit will drive "a big flash tube like the ones used in cameras." This tells us nothing. Flash tubes cover a huge spectrum and so do cameras. How many joules per flash? Duration of light output in microseconds? Rise time required? Does it require synchronization to an event or a series of events?
You provided no information on how you intend to trigger your flash lamp(s). Connecting the charged capacitors to the flash lamp(s) by means of contacts on a relay is not advisable. The usual way to trigger a flash lamp is with a high-voltage pulse, either capacitive coupled to the tube, or by means of a pulse transformer in series with the flash tube, while the tube is connected to the charged capacitor(s) that provide the energy for the momentary flash discharge. Perhaps your relay is intended to be a safety disconnect to allow the flash lamp to be handled? If so, the normally-closed contact on a Form C relay (not a Form A as shown in your sketch) should be used to discharge the capacitor bank through a resistor.
For a specific design, a lot depends on exactly which flash tube(s) you plan to use. I would ask the manufacturer for information on how they recommend their flash tube be triggered. You also need to consider how often you need to trigger the flash tube and how much energy the tube will dissipate after triggering. These two factors will determine how much power your circuit must deliver, hence the supply voltage, transformer turns ratio and core composition, the transistor power ratings, and the output voltage and capacitor values needed. The flash tube manufacturer should be able to provide information on the hold-off voltage, the maximum discharge energy, the maximum pulse width, the maximum pulse repetition rate, and what cooling (if any) is needed.
Really large flash tubes require cooling, either forced air or by immersion in a circulating fluid with a heat exchanger and refrigerated cooling. For example, some solid-rod lasers are mounted in an elliptical cavity with the flash lamp at one focus and the laser rod at the other focus. De-ionized water, kerosene, and 3M Fluorinert FC-40 have all been used in the past for liquid cooling because they are non-conductive as well as transparent, but you are free to think of something else if liquid cooling is necessary. If operation in free air is your plan... then never mind.
Perhaps these questions mean nothing to you. Maybe they are irrelevant to what you are trying to do? If so, please pardon my intrusion.