Lucette from Cinderella Phenomenon comes to mind! I wouldn't necessarily call her a "loser," but she definitely fits the socially inept part. She's blunt, distrustful, and pretty bad at connecting with people, even when they're genuinely trying to be nice to her. A lot of her attempts at interacting with others end up coming across as rude or dismissive because of how she was raised.
What I like is that the game actually lets those traits be flaws. People don't automatically adore her or find her attitude cute, and she has to put effort into understanding others and recognizing when she's wrong. Watching her slowly learn how to let people in is really satisfying.