I've always chosen the 'she is more than she thinks and will find another way to help her family' options when talking to her in her friendship events, and have always had her on my (winning) team in the ship race (Hamin + Cordelia + Penelope + Lyon or Avalie), and have always been able to get her to go to Skalt in the end!
I think you might need to boost her confidence during the play too? I had a Sheltered Princess playthrough where she played the character Penelope usually does, and Penelope was too scared to go to Skalt. But it could have been because my persuasion or some other skill wasn't high enough to convince her.