It looks to me like the coauthorship decision is being clouded by the graduate student supervisor's annoying behavior, including his claims of being "an equal author/developer". Whether this graduate student should be a coauthor is not a part of the question -- rather the question is how "to get rid of" him. But based on what you've said I think this is wrong: you do need to consider the case for his coauthorship. To my mind it rests on two things:
1) He was assigned at the beginning by your PI, whom you say absolutely did supervise you and was crucial in the creation/implementation of the model, and whom you will be including as a coauthor. Well, part of your PI's supervision was to assign this graduate student to you, whom you met with much more frequently than the PI. Thus the three of you entered into a collaboration.
2) There seems to be no doubt that the graduate student followed through with the process of supervising you. You write:
Supervision consisted of meeting once a week to update him on my progress so far. Sometimes he would provide certain helpful suggestions (none that are actual innovations just slight fixes/alterations that would improve the model).
So he met with you regularly -- more regularly than your busy PI. Regular weekly meetings are amazingly helpful in keeping people on track (especially at the junior level...but also at the senior level, honestly). He didn't just listen to you but provided helpful suggestions. And not just suggestions that sounded helpful but some which actually improved the model. Thus he made an intellectual contribution to the work.
The confluence of 1) and 2) makes your desire to have the graduate student supervisor not be a coauthor look unreasonable to me. Coauthorship is a convenant that people enter into: it is an agreement that they will do certain work and as a result be part of the final product. There is a certain base level of involvement that various professions and journals require for coauthorship: that seems to be safely met here. Collaborators are also free to impose higher standards, but these standards should be made clear in advance. It is very uncollegial for you to turn around after work has been done of the form that was specified and try to shut someone out of coauthorship.
In general, I would say that if someone does what they were asked to do procedurally for coauthorship but comes up a bit short intellectually -- i.e., it turns out in retrospect that their contributions are not so valuable or essential to the final paper -- then the decision on whether to withdraw from authorship rests with them and not the other collaborators. If you feel that someone else didn't pull their intellectual weight, then the time to bring this up is in a discussion of whether the collaboration should continue. (I should say that most people I know have a very acute sense of "not pulling their intellectual weight", and it is rather rare to see a math paper with a coauthor who could not point to a theorem or proof in a paper and say "I did this part". But other fields may differ.) This is still a delicate conversation, of course.
I think what you are really trying to say is that you want to be first author. Based on your description of the work, it sounds reasonable that you would be either first author or co-first author with the PI. That is a discussion for the three of you to have.
I would like to achieve this in such a way that does not negatively impact my relationship with my PI, as grad schools require reference letters.
Yes, be careful about this. I am going to guess that the PI will not be pleased at an attempt to cut out his own student from the paper: that is going against the plan for the work that he set up. Finally: "...as grad schools require reference letters". Hmm. True gratitude is golden, but knowing which side of your bread is buttered has got to be worth something.
