Note that the only item that uses 3ph power is the spindle motor. (on some models the coolant pump might be 3ph)
Personally I would make some space in the left side cabnet by moving or eliminating contactor units and mount a VFD there - The VFD can then handle direction and even tweak your top and bottom speeds. Also using VFD breaking will extend the life of your mechanical spindle break.
This method is "cleaner" and will use less power than a constant on motor type phase converter.