Hi Dane, not a local but been there 3 times.
Since you've researched the offical gems I'll spare you my thoughts on those.
Once you are in BCN please consider to travel to Valencia. This is a truly hidden gem among the big spanish cities. It's a short train ride and you'll see another beautiful city - in my point of view the most stunning one (Madrid, Barcelona, Valencia).
Here you get an idea (it's in german, but google translate will cover you)
https://21hemoon.com/valencia-perle-am-mittelmeer/
Re your train ride to Paris. Not sure why not use the plane because it'll save you time. #my2cents #travel #Europe #barcelona #valencia