Can I send a letter to the editor and get my name published on the paper if my name was published on the abstract but not the final accepted manuscript. How should I best handle this sensitive situation.
If it was the journal that made an error then they can correct it, and you can ask. But that seems very unlikely. It is more likely that the PI that submitted the paper made a change in the text of the document submitted. That may have been either a choice or a mistake. If the article was published as submitted then the journal isn't likely to change it without permission of the submitter.
If it was a choice, and you have to ask the PI to learn if that was the case, then you are due at least an answer as to why. If you dispute the answer when given then you can defend your authorship position to the PI and (see below) to the journal.
I would guess that you will have to eventually speak to the PI. But you can start with questions, not demands.
In a serious dispute, which isn't good for a student to engage in, you can demand that the journal retract the paper if you have a valid claim to authorship. Such things rarely turn out well and fighting with a PI, especially a supervisor, is seldom satisfactory in its outcome. You probably cannot demand that the journal add your name without the permission of all authors, including the PI.