To Pearl it was just a shiny thing. A shiny thing that would save her sister's life, so of course she would be trying hard to get it. She was unaware of what it was or what it did or that it would give Eliza her legs back. She didn't make any deal with MC, nor agree to help him get it, so therefore by the very definition it cannot be a betrayal.
If you and I both want the last piece of chocolate and you take it first, that is not a betrayal (unless you agreed to let me have it). I'll be pissed at you, but you haven't betrayed me.
You can hate her cause she beat you to it, you can be pissed that her actions put Eliza's recovery at risk, you can dislike her as a person, but you cannot claim she betrayed you.
If MC had told her what he needed it for, knowing what we know of Pearl, they most likely would have agreed to use it to heal Eliza and then Pearl can take it back to save Kira. Everyone wins.
It's also quite interesting that Selena, Kira & Mei, who all knowingly put MC and/or Eliza lives in mortal danger, show no remorse or offer any real apology for their actions. While Pearl, who didn't know what she did, carries the guilt with her for ages and is extremely apologetic.