Seriously consider Rotterdam to Munich and then through the Brenner Pass to Bologna and then down to Rome. An overnight in Munich and a morning ECB to Verona or Bologna then a regular Frecciarossa will get you to Rome about 6pm.
Rotterdam to Munich has many routings but one of the least complicated is R’dam to Utrecht, on to Dusseldorf and the ICE to Munich. Avoid Thalys connections - they are very expensive for relatively short journeys. The ECB trains have a 10 euro supplement but reservations are included if you buy from OEBB. Frecciarossa reservations (also bookable from OEBB) costs 10 euro.
Have you checked suitable options on the rail planner or on www.bahn.de? That’s where I would start.
It’s possible in a long day via Paris, but with expensive extra supplements. I would probably go via Germany and Switzerland. You could look at the night train from Amsterdam or Utrecht to Basel as a start.