I would suggest the 'side to side' vibration was ground resonance rather than dynamic rollover.
I had this very badly in a Wessex many years ago on rotor start due to unequal tyre pressures (servicing error) and by the time I retarded the speed select lever and applied the rotor brake each wheel was alternately lifting over a foot off the ground and the ground crew were heading in the opposite direction very quickly.
Fortunately we didn't rollover but it was a scary moment.
It can be a function of tyre pressures, oleo pressures or a MR imbalance aggravating the natural frequency of the undercarriage.
The sequence of events described in the reports matches exactly an uncorrected ground resonance situation - eventually the rapidly increasing magnitude of the 'padding' would cause a rotor strike.
The solution is to lift to the hover if you have flying Nr.