"Don’t use the output of the converter for control purposes.
Use the input voltage"
Forgive my ignorance please. Are you saying the magnetic switch should be on the 220 single phase input side of my converter?
Not really. The power for the magnetic switch should be drawn from the single phase source. The mag switch has a coil that pulls in the main contacts. That coil needs power. Make sure that coil has nice clean power, at the voltage that the coil is designed for. It is very common on machine tools to use a control transformer connected to two terminals of the input (whether single phase, or three phase), then the machine control is done with this reduced voltage (often 120v, sometimes 24v). Some simpler setups will use line voltage for the control power, so no control transformer (I have a table saw like that). The point is that it needs proper control power. Do not take the control power from the added leg out of the phase converter.
1. Check what voltage the mag switch coil is designed for (perhaps a name plate, often a label on the coil itself). If it is rated for the line voltage, you are good.
2. Have a good look at the small wires that connect around the mag switch. They will connect on L1 and L2, or L1 and L3, or L2 and L3. (Perhaps you just need to swap two of the mag contactor input wires). Then you may have to swap your T1/T2/T3 wires around to get the correct motor rotation.
3. You ought to check the current draw on each leg, so as not to be running above the motor nameplate (don’t just crank up the motor overload sensor to see what happens - one of the options is that the magical smoke will come out of the motor - really hard to put it back in).
Let us know how you make out.