After learning that North America made 2 lamp ballasts for mercury vapor and probe start metal halide lamps where both lamps are operated in series, I am wondering how I can go about adding superimposed ignitors to mercury vapor versions of such ballasts with an OCV between 480v-515v in order to run 2 pulse start metal halide lamps on these ballasts? I drew out a diagram where 2 220v-240v superimposed ignitors are used where one is used for each lamp. I am wondering whether I should use a single 380v-440v superimposed ignitor or 2 220v-240v superimposed ignitors in which one is used for each lamp?

That won't work and very likely it will kill the ignitors.
The problem is, the upper ignitor exposes the input of the 2nd one to HV and the top lamp current has nowhere to go after ignition, so it won't be successful.
The 2nd one, beside getting HV pulses from the 1st one via the lamp, has no supply, so wont be able to generate anything.

For such series connection you really need one ignitor, designed to ignite the whole lamp series chain, so mainly double the arc detection threshold voltage.
The peak ignition voltage does not have to be that high, because during this peak the capacitance from the net between both lamps to the ground makes sure the top lamp sees full peak voltage first, its gas breaks down and so passes the high voltage across the 2nd lamp.
So you would need a custom ignitor. Or know the exact schematic and component values of one single lamp ignitor, so you will be able to design an extra impedance put into the N wire of the ignitor to raise its threshold voltage.

That won't work and very likely it will kill the ignitors.
The problem is, the upper ignitor exposes the input of the 2nd one to HV and the top lamp current has nowhere to go after ignition, so it won't be successful.
The 2nd one, beside getting HV pulses from the 1st one via the lamp, has no supply, so wont be able to generate anything.

For such series connection you really need one ignitor, designed to ignite the whole lamp series chain, so mainly double the arc detection threshold voltage.
The peak ignition voltage does not have to be that high, because during this peak the capacitance from the net between both lamps to the ground makes sure the top lamp sees full peak voltage first, its gas breaks down and so passes the high voltage across the 2nd lamp.
So you would need a custom ignitor. Or know the exact schematic and component values of one single lamp ignitor, so you will be able to design an extra impedance put into the N wire of the ignitor to raise its threshold voltage.

So, in that case, I might need a single 380v-440v superimposed ignitor for striking both pulse start metal halide lamps?

So, in that case, I might need a single 380v-440v superimposed ignitor for striking both pulse start metal halide lamps?

You need an ignitor, which will fire on the OCV there, but won't fire on the sum of both arc voltages, including their reignition overshoots.
So probably needs trigger voltage around 400V after the RC filter (single pulse MH ignitors use about 10kOhm charging 100nF and then a 200V SIDAC discharging the capacitor into the primary of the pulse transformer; or into the tap of the ballast winding, which then is acting as the pulse transformer).