I think someone already suggested that as an option on an earlier
thread:
Agnes, yep, that has been suggested several times, including this other thread this morning.
https://community.ricksteves.com/travel-forum/italy/transportation-rome-to-sorrento
But as I noted in that thread, it wouldn't give me a search result for Aug 15th. That could be because - as Dario noted in the other thread you'd linked - Aug 15th is a holiday. Flixbus did appear to be running on that day, though. Yes, it takes 4+ hours but I'm not sure a driver from Rome would be significantly faster. I'm sure someone else has answer for that...
Still, a fast train can get from Rome to Naples in a little over hour.
Edting to add: so you've only asked about Rome>Sorrento. Where are you going after your stay in Sorrento? Just curious as often trippers heading south have to head north again to fly out of Rome. Or something.