Train from Monterosso to Genova for a flight

How long does the bus take from the train station to the airport--have you checked its schedule?
And where is the flight going--within Europe or Intl? Checking bags? Online check-in?

You still have to get from Genova P. Principe to the airport. There is an express bus or taxi, but it is still going to take time. Figure at least a half hour, so now you are arriving for a 9:45 flight at 8:45.
Thinking the 5:17 IC train might be better. Arrives at 6:41, making a comfortable transfer to the airport.
That train originates in La Spezia and Monterosso is the 2nd stop, so not much chance that it will run behind, at least at the start. 3 hours is a decent cushion if delays pop up.

I prefer to sleep in my departure city the night prior to my flight.
I wouldn’t want to miss my flight due to some unforeseen delays.

Last October I was on a train enroute to Rome where I met a couple, from Australia, who almost missed their flight because the train arrived nearly an hour later than expected.

Enjoy your trip!

I prefer to sleep in my departure city the night prior to my flight.
I wouldn’t want to miss my flight due to some unforeseen delays.

Ditto!

I would consider splashing out for a private transfer, should run around 100 as a wild guess (for a one hour taxi ride, but peace of mind).

Would you feel comfortable taking the 6:45am train arriving 8:15am in
time for a 9:45am flight?

Short answer? Nope.

Would you feel comfortable taking the 6:45am train arriving 8:15am in time for a 9:45am flight?

NO, NO!, Absolutely NO!!

Someone would have to have a really big gun pointed at my head to make me try something like that