'Unique' experiences in Barcelona

"Unique" is a pretty subjective term, but I notice you didn't include Saint Pau in your list of sites. It's a short walk from Sagrada Familia, with maybe 10% of the crowds. Montaner isn't as famous as Gaudi, but I like him better.

https://www.catalunya.com/recinte-modernista-de-sant-pau-17-16003-300?language=en

Visiting Nou Sardenya stadium and watching a CE Europa football (soccer) match: http://www.ceeuropa.cat/english/in-english

Visiting Monasteri Sant Pau del Camp:
https://www.barcelona.com/barcelonadirectory/monuments/monasterisantpaudel_camp

Most 'unique' experiences in Barcelona happen on a certain day at a certain time in a specific place. You haven't given us enough information ("after Easter" is too vague) to make suggestions. For example there could be a local fiesta or castellers event during the time you're planning to be here. Without knowing your precise dates it's difficult to suggest 'unique' sights or experiences. Enjoy your visit.

Had to chuckle at (the very useful and highly recommended suggestion) of what's now known as Recinte Modernista de Hospital Sant Pau: https://santpaubarcelona.org/en/recinte-modernista/qui-som/.
Recall taking a journalist for Forbes magazine (a family friend) there when it was still a functioning hospital — she was (as we say in Britland) 'gobsmacked'. And next Tuesday I have an appointment to see a consultant at Fundació Puigvert, a modern functioning hospital cleverly bullt on the footptrint of an old hospital Sant Pau pavilion and which still contains many original modernista details.

You might enjoy a Devour tours food tour. They blend history, culture, and you test different foods all around ours began at a wonderful market it was a great way to learn what we’d want to eat more of as well as getting history and an enriched few hours of our day.

We're also visiting Barcelona in April, so I can only share our ideas, not actual experience. BUT, we're watching La Catedral del Mar on Netflix with the intent of visiting the church (Santa Maria del Mar) on our trip. We're hoping the connection to the miniseries makes it all the more interesting. Then, practically next door, is the Museu Frederic Marès. From our research, it contains an outstanding collection of medieval sculpture, going back to the 3rd and 4th centuries. The top floors are a wide collection of 19th century ephemera and objects. Like I said, these are untested suggestions, but we are greatly looking forward to the day...

The Mares museum is quirky and worth a visit if you want to avoid the more obvious options. The Picasso Museum can be unbelievably crowded (think of the photos you may have seen of the Vatican Museums), so I recommend it only to serious fans of Picasso. The Miro Museum and the MNAC were very pleasant art venues when I visited them in 2016. They are not undiscovered, however.

The Caixa Forum, an arts/cultural center, is located in a repurposed modernista factory building.

You should consider walking back downtown through the Gracia neighborhood after you visit Parc Guell.

The Ramblas is not worth a special effort. You'll probably find yourself on it once or twice, because it's a main road, but in my opinion there's nothing special about it except the presence of pickpockets. I have no clue why Rick still recommends it.

In addition to experiences on your list, we visited Palau de Música Catalana. We enjoyed a guided tour. Both the exterior and interior are breathtaking.

Ride the cable cars up to Mont Juic then walk down and stop at the Miró Museum. A lovely museum with some wonderful views and art.

Have a coffee or hot chocolate with churros at Xurreria Laietana https://www.tripadvisor.com/Restaurant_Review-g187497-d2216792-Reviews-Churreria_Laietana-Barcelona_Catalonia.html

I can personally recommend:

Parque del Laberinto de Horta: A 18th-century neoclassical garden on a former palatial estate. It's not very well known with tourists but is very beautiful with numerous sculptures of Greco-Roman mythology along with a number of fountains, springs, and pools. In the center of the gardens is an English style hedge maze.

Hospital de Sant Pau: A Gaudi-esque modernisme complex built in the early 20th century as a hospital. It's about a 10 minute walk north of La Sagrada Familia. It's now a museum and has been named a UNESCO World Heritage Site. Sant Pau has significantly less tourists than La Sagrada Familia.

MUHBA Plaza del Rey underground Roman Ruins of Barcino

Tibidabo the oldest amusement park in Spain plus the best views of the city.

RE the Marès Museum:
"Sabine Hassell turned into the courtyard of the Marès Museum. After paying for her ticket she turned from the desk to put her backpack into a locker. She caught sight of a man silhouetted in the arched entrance. Late twenties, medium build – she vaguely recognised him as a fellow train passenger. Their gaze connected. He dithered and opened up a streetmap, and stepped backwards into the busy alley. She turned away and entered the museum.
The Museu Sentimental, situated upstairs above a collection of crucifixes and Madonnas, is a whole life’s collecting of everyday ephemera arranged under glass. Marès, she learned, was a sculptor who designed Franco’s victory monument on Cinco de Oros, and an avid collector of useless things: tram tickets; fans; dolls; toy theatres; tooth-picks; printers’ handbills; menus; napkins; cigarette papers; cigar bands; books of matches; walking-sticks ... and it goes on and on and on and on…."
An extract from the novel, After Goya

Thank you everyone for the wonderful suggestions. There are several here that I will be looking into. I truly appreciate the feedback and value your experience/expertise!