This isn't an answer, just some points you can consider and some 2 cents:
Given the short time, if I were in your shoes, I'd:
Comb through the student academic handbook about sick leave and how should the makeup examination be scheduled. Look for keywords like "... upon mutual agreement." or "... student and department will decide." etc. and quote that in a written document to urge the committee to honor the wording in student academic handbook.
Identify the chair of the grievance committee. Every department/school should have one, and inform him/her about your situation. Ask for guidance and potential on filing a grievance.
Objectively figure out what are the number of quality days available for a non-sick student to prepare for the tests, and then come up with how many quality days you have been given, minus the ones that you have spent. Ask for the exam to be rescheduled to a reasonable time that would provide a fairground to you. Down play the fact that you "want more time," emphasize on the desire "to be judged fairly." Should the number of days are actually comparable, chance is you really have no ground to delay.
Chat with the admission people and discuss the possibilities on the followings: i) retrospectively request a medical leave (this is more viable if you are on a trimester system or this is a half-credit course, missing 4 weeks might as well mean you have not taken it at all) for this semester, and retake the whole thing again. ii) Negotiate for fee reduction or fee waiving so that you don't have to pay again. iii) Also make sure the half-done course will not end up as F or even appear on your transcript.
If you think you're not being fairly judged, never take the exam. Once a grade is given it's next to impossible to revoke. Inform the school as soon as possible with your reasons (preferably sound and focusing on the lack of fairness), give them a summary of your arguments and what you have talked about with different representatives in the department.
I think, in a sad way, that you only decide to decline one day before the rescheduled date is an extremely bad move. If indeed everyone also had one week to prepare, then it is pretty much done for. When the decision was just made, you can still negotiate for 2-4 more days to collect lecture slides, notes, and meet with the lecturers. Now, it's a bit too late to do that.
In a nut shell, set your worst scenario and figure out what are the things that you cannot afford to mess up. If your first thing is GPA, then strive to make this trimester/semester not counted. The cons: you may need to retake, but given your circumstance, retaking seems a lot more viable than taking an exam. Also, don't position yourself as a victim. It's unfortunate but you're not asking for leniency or special treatment. Show them that you're very conscious about the grade situation and your determination to work through this glitch with the department, and hopefully they will reopen the case.
I wish you best of luck, and if you have time, come back to update us by revising your question.