It's because the NPN won't switch correctly when its collector is negative with respect to its emitter.

 

Is this because the gate voltage for the triac should be alternating also?

Look on the data sheet for gate trigger quadrants.

Taking the gate negative works best most of the time, but it should be possible to trigger a triac with any permutation.

 

It's because the NPN won't switch correctly when its collector is negative with respect to its emitter.

But when the collector is connected to the 5 V rail, it will never be negative with respect to the emitter?

What about three 39 V zener diodes in series to snub the voltage when all lamps are unlit?

The MOC3083 has Critical Rate of Rise of Off–State Voltage 600 (min)-1500 (average) V/µs - is that relevant to the frequency it can operate at?

 

But when the collector is connected to the 5 V rail, it will never be negative with respect to the emitter

True. But you can't drive the transformer with 5V DC as shown.

What about three 39 V zener diodes in series to snub the voltage

Ok perhaps for DC, but not AC. You'd need two series strings of 3, or one string of 3 across a diode bridge.

is that relevant to the frequency it can operate at?

Yes. If the inverter frequency is, say, 20kHz and if it has a peak voltage of 1kV then if the voltage rose linearly its rate of rise would be 1000V/50uS = 20V/uS. That is well below the critical 600V/uS, so should not cause false firing of the triac. But that's a lot of 'ifs' .

 

I have improved the design again wherein the value of the 1M resistor has been changed to 510R. I also read of a component called IL420 which has a high "critical rate of rising voltage whilst commutating" (70 times that of the MOC3030). The cost is about twice that of the MOC3083 and the maximum voltage rating is 600 V. They do a 800 V version called IL4208 but that is about five times the cost of the IL420.

If the IL420 doesn't work, maybe it would be feasible to make a small inverter for 5 V to 90 V peak AC. It only needs to be like 50 mW of power but it also needs to limit the current somehow to ~1 mA. I guess a capacitor would do that.