Thank you for all of the information. The cheapest flight is only ~$5 less than the SLC-AMS-FCO flight and it has a 7hr layover in JFK. I will go ahead and book that flight and then watch for the schedule changes and see if I can move into something better as time goes along.
Just so I am clear about the process in AMS...we would deplane and go through passport control (which has an express lane for tight connections), but maybe/maybe not security? Our bags would be transferred to KLM seamlessly in theory and we would go through customs in Rome.