I have much sympathy for you feelings, anon. Testing and ranks are often much overemphasized. I was myself a difficult student who was reluctant to take grading tests, not taking grading tests even when the main teacher of the dojo told me I should. (In that dojo, tests are given a couple of times a year and you decide yourself if you want to try or not - the main teacher only pushed those he felt lagged behind in the ranking scale.)
Anyhow, in this weird little aikido world ranks and tests do exist. One has to adapt to one's environments, or life will be very difficult. Then of course is the question how much one needs to adapt.
Can I respectfully request that I not take the test?
If I do have to take the test, can I respectfully request that I may continue to wear my white belt?
You can ask not to test, of course. Then is the question how your teacher will respond to it. You can ask if it is OK to wear a white belt even if your rank in your dojo usually wears another colour. My guess is your teacher will tell you to join the line and do like everyone else, i.e. take the tests when he tells you to and wear the rank you have. There is a system to it, and people who do not follow the system is a disturbance. "A nail that sticks up, will be hammered down".
Probably not in your dojo (since your teacher tells students when they should test) but in some other places it happens that individuals stay on kyu rank forever simply because they do not want to test. Around here belts are either white or black, no colour, so they stay white belts forever. People who do the same thing often have their individual reasons that might be all different, but one aspect of it can be.. a kind of reverse snobbery, which is a kind of snobbery in itself. I remember looking down on people hurrying up the rank scale. Later I realised that if I could stay on a lower rank and not look down on those who tried to take as big chunks of the pie as they could, the way they saw fit - then my take on it would have been OK. I could not, and if staying on a lower rank made me despise people I should change. At this stage I went back to accepting to take the grading tests.