The purpose of punctuation is to clarify what goes with what, where independent ideas begin and end, how elements in a series are to be understood, who said what, and more. But if punctuation only creates questions about what the writer has in mind, it isn't serving any useful purpose. That, I think, is the problem at the core of your question.
You start with a sentence that contains a question enclosed in parentheses, and you want to place that parenthetical question at the end of a sentence that asks a question of its own. And then you want to know where to put the question mark or marks.
This, I think, is the wrong way to approach a question about punctuation. The central question about punctuation that a writer should try to answer is, How do I punctuate these words so that my meaning is immediately clear to the reader? And in the case of your example wording, the answer is, First get rid of the parentheses, and then figure out the best punctuation to use.
In my view the simplest way to make the meaning of the example wording clear is by adding an em dash before the start of what had been the parenthetical phrase:
Hi, John. Could you please help me get the account access described below—or help me route this request correctly so it can be fulfilled?
The em dash signals that you are breaking away from the initial question to ask a different but related question. The result is a straightforward presentation of the writer's request to John that enables it to proceed without resorting to double question marks, unnecessary parentheses, or other distractingly elaborate punctuation.
A less forceful approach would be to replace the em dash with a comma:
Hi, John. Could you please help me get the account access described below, or help me route this request correctly so it can be fulfilled?
This punctuation seems a bit mild to me, given the rather sharp shift in the question that the wording implies, but some writers prefer commas to stronger punctuation marks. Ultimately, there is plenty of room for writers to differ in their punctuation preferences without adopting a form of punctuation that supplants the words and their meaning as the center of attention.
