There should be no "heavy fines" for 2mm over.
In fact they don't "fine" you at all. Any additional charges are related to "what it would have been if calculated correctly" plus potentially a small processing fee.  So no, 2mm over will NOT trigger "heavy fines". 

They round up to the nearest cm so yes 2mm over could round up to another cm, and that 1cm more could tip it into a new price bracket if it goes over a certain cubic size. But that's just a cost recouping, "heavy fines" implies there is a crime here, which is not the case. It's just you being charged what you should have been charged if you had put the measurements in right.

Now does this $129 or whatever it was from the OP sound reasonable? Probably not. But without knowing what the parcel size and weight they put into the system, what the actual parcel size and weight was, if it went from a flat rate to a calculated rate, and where it was going from and to, we can't know. On the surface it sounds high and they should challenge it, because the scanning systems DO make mistakes sometimes. But either way it's not from "heavy fines"