So, a couple of things to bear in mind. First, most of us (maybe all of us) are more insulin resistant in the morning than the evening, plus we get a nice glycogen dump from the liver (also called Dawn Phenomenon). This combination means that the same amount of carbs eaten for breakfast are likely to have a much larger effect on BG than that amount eaten for lunch or dinner (especially if you've already exercised that day). Second, most non insuline-dependent Type 2s still produce significant amounts of endogenous insulin in response to eating carbs.
Put those together, and that is a perfect prescription for what Brian brought up: reactive hypoglycemia. That doesn't mean what is happening to you is RH, but if I were a betting man I'd put money on it. Do the coffee experiment, by all means. But definitely try Brian's suggestion as well (skipping 35 g of carbs at breakfast certainly can't do you any harm): try eating eggs and cheese or whatever else works, but keep breakfast to less than 5 g of carbs for a few days. Don't do it at the same time as the coffee experiment (confounding effects), but do it after you've satisfied your curiosity on that front.
As for the original question, the answer for me is "No, coffee doesn't affect my BG one way or the other," although the milk I sometimes put in it can.
To expand on my "carbs at different times of the day" thought, my own experience is pretty straightforward:
I generally have less than 10g of carbs before noon. I'll often have 30g for lunch, and 60g for snacks and dinner (usually after I've worked out or done a lot of walking). For some reason that 10-30-60 distribution means most days my BG is pretty flat from fasting through the day, with my lowest point usually coming before dinner. Everyone is different, but my pattern isn't particularly unusual I don't think. It has to do with glycogen storage in the liver, the way muscles use glucose after even moderate exercise, and etc.