I can't say why other people are not answering this question, but I can say why I wouldn't attempt to answer this question. It's for two closely related problems:
- Too much confusing detail: There are a lot of details highly specific to your university's policies, which are difficult to understand and keep track of. If we actually need to know all of those details to answer your question, then it's probably too specific (see "Can I ask about my specific situation?" in the help center.) If we don't need to know all these specific details, then they are confusing and somewhat offputting to potential answerers. (And there are some details we definitely don't need to know, like the details of the specific preliminary exams.)
- Lack of general, easy to comprehend description of your scenario: The general question you ask is "How do I ask for a change in administrative graduate pay raise politely to my graduate department advisor with optimal chances of success?" In order to understand what you mean by "administrative graduate pay raise," I have to wade through all the details I mentioned above, because there's no tl;dr description of what you mean.
It might be better if you replaced all the detail with something like this (I don't know if I'm representing your situation accurately, because I find the details somewhat confusing, but this is my best shot):
tl;dr Because I followed a "non-standard" accelerated path through preliminary exams, I am not eligible for any of my department's standard summer funding options. How can I politely ask for a pay raise to help me cover the summer?
Specifically, in my department, graduate students who have not completed preliminary exams are eligible for "prelim funding" in the summer. I am no longer eligible for this funding, because I passed my preliminary exams early. Students who have passed oral exams and completed a certain number of credits are eligible for extra pay in the spring semester to help cover the summer, but even if I pass my oral exams I won't have enough credits in time to qualify for this funding (again, because I am "ahead" of the standard path at my institution). My research advisor has informed me that there is no research funding available this summer, and he has no further advice for me.
The graduate school has informed me that my department has the right to change pay level policy under certain conditions. I have scheduled to meet with my graduate department advisor on Monday. How do I ask for a change in administrative graduate pay raise politely to my graduate department advisor with optimal chances of success?