I think no no is a good answer. I certainly won't be putting it in the dishwasher. Although it might survive, after spending all that time turning it, do you really want to take a chance?
You, my friend, have a serious fixation on that dishwasher. Since I just got a dishwasher for the first time, I understand some of the fixation.:d I have a solution that I hesitate to suggest, but here goes:
The insert does have a threaded nut on the bottom. You could turn the mug, finish both inside and outside. Then turn a "plug" for the bottom and use a countersunk bolt to keep the insert on the mug. You could then handwash and if particularly messy, remove the bolt, and disassemble for a trip in the dishwasher.
I hesitate to suggest this because your inside turning then has to be good enough to be viewable in addition to the outside. I think this could look really sharp and might be a selling point.
