Epic LT pilot reported systems failures and a failure to recognize a configuration issue caused a loss of control during landing; causing a propeller strike.

Narrative

The incident occurred during the final landing of a three segment; daytime trip involving actual instrument conditions. The en route portion was conducted at FL340. Descent to landing would require descent through a region of moisture above the freezing level on the upwind side of the ZZZ1. ATC was queried prior to the descent for any active icing reports with none reported in the area of the flight. The approach over the mountains and into the sea level airport would require multiple step down altitudes increasing the time spent in potential icing conditions. Pitot heat is routinely used in this aircraft when operating at high altitude whether or not there is visible moisture. During the initial descent a fault was detected in the inertial separator mechanism and an error message displayed to the pilot on the Crew Alerting System (CAS). As per the PIM the circuit breaker was pulled rendering the system inoperative. The temperature at the airport was +19 C and any inadvertent ice accumulated during the descent was expected to dissipate. As a precaution propeller heat was turned on prior to entering the clouds. AWOS at the destination airport was monitored several times during the descent. The last report included wind at 210/6 kts; 4500 ft. scattered; 7000 ft. overcast. A request was made for a possible visual approach along with a fallback RNAV approach. The aircraft exited the cloud deck around 6400 ft.; but the lower level scattered clouds obscured portions of the ground and airport initially. At 5200 ft. the conditions were good enough to allow for VFR continuance and cancellation of the IFR clearance was made prior to descending into the airport traffic area.The aircraft had been appropriately slowed with take-off flaps selected and the prelanding checklist run during the descent in clouds. There was no traffic in the area at or below 5200 ft. and altitude was lost by performing a series of turns with the autopilot disengaged while extending the gear and selecting landing flaps ending with proper positioning for a 2 - 3 mile final approach. The final approach was stabilized with an IAS of 95 - 100 kts. Over the threshold power was completely reduced and a round out-to-flare was commenced. As the nose started to pitch up an aural warning of an impending stall started followed rapidly by an aural 'Push Push' with stick pusher engagement. The nose pitched down and the nose gear wheel barrowed with continuous stick down pressure which could not be overcome by the Pilot in Command (PIC). The PIC held down the AP/Trim disengagement switch and expected this to result in disengagement of the stick shaker/pusher. When the stick pusher stopped the back pressure exerted by the PIC on the control column led to ballooning and the aural 'Push Push' with stick pusher engagement immediately started again. The PIC re-asserted the AP disengagement switch which did stop the process from occurring a third time and the final porpoise was controlled manually without further incident.Reverse thrust was used after stabilization on the ground; the aircraft handled normally on taxiing all the way to its regular tie down spot over a mile from the runway turnoff. Following a normal shutdown sequence with stable engine instrument indications; inspection of the propeller demonstrated limited damage to the distal tips of the composite blades. After securing the aircraft the runway was walked and closely inspected for evidence of damage or FOD neither being found. Discussion with the manufacturer's representative after the incident pointed to the use of propeller deice automatically engaging the ICE PUSHER MODE leading to the strong stick shaker/pusher action when the aircraft is slowed to what would otherwise be a normal approach speed. The rapid unexpected sequence of stick pusher engagement at a probable 5 ft. or less over the runway during round out was the proximate cause of the incident. The commanded pitch down was exacerbated by the deployment of landing flaps during the last phase of the circle-to-land while failing to note a CAS message which would have announced the ICE PUSHER MODE engagement. While the PIC thought that he had managed to stop the first stick pusher sequence by holding the AP/Trim disengagement switch the required 2 seconds; this proved not to be the case. The secondary incipient stall caused by the initial automated disengagement of the stick pusher led in turn to re engagement of the system and a second porpoise which the PIC was not able to overcome in time to avoid a repeat of the nose wheel first contact. The final ballooning was controlled by the PIC without significant difficulty after he had managed to disengage the system.The entire sequence of events leading up to the incident started with a concerning distraction (INERT SEP FAILURE) during descent into possible icing conditions. The CAS message warning of the automated ICE PUSHER MODE would have occurred proximate to the final phases of the flight during which additional distractions contributed to increased pilot workload: 1) the series of step down altitude clearances while dealing with mountain turbulence/airspeed variations; 2) the combination of an overcast with lower scattered clouds leading to a position where the aircraft needed to lose additional altitude close in to the airport in order to complete a visual approach; 3) the rapid transition from an IFR descent with potential icing to marginal VFR conditions followed by the final visual approach. While the aircraft is not currently certified for flight into known icing conditions it has a complete suite of functions needed to achieve icing certification including the automated ICE PUSHER MODE. Ground school includes instruction on the proper configuration of the aircraft when the ICE PUSHER MODE automatically engages; but there is limited additional discussion of the anti-ice systems for context since the aircraft is not currently certified. No further specific training in these features was offered in the simulator or in the actual aircraft.The ICE PUSHER MODE system clearly exists for an important reason; but akin to other automated systems (e.g. Boeing MCAS) formal instruction in and introduction to recovering from an inadvertent triggering are very important for the new trainee. Increased emphasis should probably be placed on the actual consequences of the ICE PUSHER MODE: an approach to landing can rapidly convert to a significant control situation during round out with little or no prior aerodynamic or aural warning. Simulator training might demonstrate the best way to recover from this situation during the final landing stages is to add go around power when the first unexpected aural warning is heard or that the application of power is a poor choice to be avoided.