Voltage transfomer without ground connection
Usually the ground connection on soldering irons are used for grounding the tip, preventing electrostatic discharge to potentially sensitive components. If the Hakko iron has this feature, and you use a transformer without a ground connection, then the feature won't work and you will endanger your static-sensitive components. You could connect the ground from the iron straight through to mains ground without connecting to the transformer, but it would be even better to have it connected to the transformer core. Then make sure you use a 3-prong plug (with a ground prong) and connect that pin to the transformer core as well. Check if your transformer is already mid-point grounded--that's where the transformer part number would come in handy.