huiyee, your problem with the CT is time: as it is you'll realistically have 2.5 days in Venice, 2.5 days in Florence, and 3.5 days in Rome: figure on losing 1/2 day every time you have a move. Another location means shorting something else, and what you have is really bare minimum to do each of the 3 cities justice.
That said, you could do it as a long day-trip from Florence. RS Italy expert Roberto very recently posted this info on a special Jubilee-year train between Firenze Campo di Marte and La Spezia that is the absolute fastest transport option.
https://community.ricksteves.com/travel-forum/italy/florence-to-cinque-terre-and-back-an-overlooked-option
La Spezia is the southern gateway to the 5 Terre, and you'd switch trains there to any of the villages. There are MANY trails within the park; you are probably referring to the "Blue Trail" that's actually 4 separate segments between Monterosso in the north and Riomaggiore in the south. Two of those segments are indeed closed and probably will remain so throughout this year.
http://www.parconazionale5terre.it/Eindex.php
Besides time, this is not a good trip for a rainy day so weather can be a complication as well. The villages are going to be mobbed, and you may feel overrun unless staying overnight - there's more breathing room when the day-trippers leave - or hitting higher, lesser-used trails.
Don't want to throw a wet rag on your interest in this but there are a number of things to consider?