A doctor's note doesn't sound like it would apply to your situation.
I'd say your best bet is to be honest and ask them to defer it. The one deferred exam, no questions asked (within reason) does seem to be a real thing. They will hopefully understand that accidents happen.
If your record is otherwise good I would say you have a good chance of them granting a deferred exam. But, don't wait! Go in ASAP, send an email today and get in there ASAP tomorrow morning.
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