You don’t need a neutral but the elements do need to be wired correctly.
You have (4) elements that are 8 ohms each. In order to have 7200 watts, you’ll need to:
1) Wire two elements in series, making two 16 ohm elements.
2) Wire the two 16 ohm elements in parallel so that you end up with an 8 ohm element.
That will be a 30 amp load at 240 volts, no neutral wire required. Changing out the 3 wire plug for a 4 wire only provides for a proper 120 volt control voltage which your modified kiln does not use (or need).
Personally, I would use mechanical relays and increase the cycle time of the controller to 30 seconds. The operating temps required versus the watt density means that turning on an element for a 30 seconds doesn’t raise the temperature by a degree, it takes several minutes to do that. Solid state relays almost always fail shorted and that results in a run away, not good with a kiln. That’s why they use mechanical controls. You could use two 20 amp contactors wired in parallel. If you’re going to stick with solid state, use four contactors, two for each 16 ohm pair to lower the amps and add some safety margin.
Proper high temperature wire is fiberglass braid over Teflon with the copper wire and all lugs being nickel plated. Anything else just oxidizes and fails at those temperatures. The right stuff is pricey and not available from the usual sources, the kiln manufacturers are probably the easiest place to source it.
My wife’s L&L kiln is controlled by their own custom PID unit that operates two mechanical contractors, one element on is low, both on is high. When the PID is within the proportional band, it use low heat, high when out, typically at startup.