Some good thoughts above about why this is an important problem to work on with him, and how.

At the same time, with an anxiety monster of my own, what is really jumping at me is the high amount of stress this is causing him, at the very beginning of every day. Which, extrapolating from my own DS, I might suspect is bleeding into anxiety about going to school at all in the morning, as well as putting him in a state where he is much less able to focus and function for the rest of the day?

If this problem is not just one moment in the day that goes badly, but something that is having big impact on his whole school day (and views of school), then I would go with your instinct that says this stress needs to be removed, completely. For now. Get him an assigned chair, show him you're listening and want to help him cope better. Once the stress is removed and his anxiety can come down, then work him him gradually to understand the nature of the block, and build up his capacity to cope with it. Reintroduce the concept slowly, with scaffolds, once he has been taught some of the skills he needs to manage the chair situation.