I understand that you cannot use your left arm, you have 'only' managed to adapt a little. Like Mario says, and myself, have your exam put off until your arm has healed.
I can imagine this exam is pretty exciting/important. Black belt is always some kind of milestone, no matter what style you are in. But I do not think it is more important than your well-being. Budo, the mother of martial arts, says self-preservation is most important.
Making the right decision is hard, but is ultimately that what builds (your) character
Thanks Tim. Over the years i have learnt my body's limitations and i will not go over the limits, no matter what. As you mention, self preservation is of most importance, esp when you have young ones waiting home for you...
With that in mind, i will go the way up till i can. If i experience discomfort or if things get difficult, i will call it a day and sit down. As it may be months before i can use the left arm again, i will try up to the point i do not cause any more harm to myself. And stay there.