No, there's no way to have a chair at your treadmill desk without some extra construction. When I need either the extra precision or every iota of concentration I can muster, I'll either stand on the frame or move to my regular desk.
I did find out last week just what a difference it's made for me, when in the midst of an insane push to finish up a report, it came time to lubricate the belt, and so I spent the next day and a half penned at my desk, and I really felt it, particularly in my knees. (Which isn't to imply that maintenance is difficult, it's pretty easy and quick, just that I'd made the decision to postpone it, and later regretted it. That whole tightening-your-shoes-in-the-middle-of-a-run thing where you think, egads! I'm not making any forward movement, rather than investing a little bit of time to make the rest of your run even better.)
Live and learn. And relearn!