WIFI thermostat no C wire
Get rid of the external transformer, and pull a proper C wire.  

It's not working because you have one side of the contactor coil connected to one transformer, and the other side connected to another transformer.  There's not a complete circuit.

When the thermostat is calling for cool, your circuit looks like this



If you follow the circuit, you can see that it's not a complete circuit.





I've highlighted the R wire in red, and the C wire in blue.