But it's not out of the way...The diversion isn't that big. SEA-FLR is 5541 miles, while SEA-EWR-FLR is 6579, an increase of less than a fifth. I've taken connections involving much more backtracking!
I think you should go via EWR or JFK. Fly whichever flight to Europe you can get from there (and there are so many that you will definitely be able to find availability for one), then connect to an AZ flight to FCO or MXP. You can then rent a car, go by train or get a domestic flight. If you can get a flight from the US to CDG, AMS, PRG or LYS, you can connect to services going to Bologna which is very near FLR. Similarly, there are flights to Pisa from CDG.
There are also some great deals for paid tickets out of NYC- under €2000 a piece (though some of these are quite inflexible and require a long adv purchase- check rules).