I would try using the Draw.DrawBegin, Map.MouseLeftButtonDown and Map.MouseMove events in something like this:
1. In Map.MouseLeftButtonDown save the current cursor position.
2. In DrawBegin detect when the shape is a circle and start tracking the cursor's movement by attaching to the MouseMove event.
3. In the MouseMove, obtain the current position and convert it (along with the center's location) to map's coordinates. Then calculate the distance.
4. Add or move a text graphic in a Graphics layer showing the diameter (distance*2).
5. In DrawComplete detach from the MouseMove event.
Not sure if you in the DrawBegin event you can access the current Cursor's location; if so you could save step 1.
Good luck