UPDATE: Oddly the problem does not exist in IE.
Amazingly I am still working on this but I do have some new insight at least. In early testing I was only looking at the IFRAME as it was being dragged and seeing that the content would disappear from it as soon as the drag began. But in more recent testing I saw that the apparently disappearing value was being preserved but within the original row (that hides from view when its clone starts to be dragged). But then when the drop occurs the original row is removed and its clone takes its place. This got me to thinking about grabbing the pertinent value at the beginning of the drag and reading it back into the clone at the end of the drop. The tricky part (for me . . . for the time being . . . ) is identifying the row that has a clone that is being dragged. If I could do it I could grab the pertinent value. But when that row has a clone being drug around there doesn't seem to be anything unique about it to target that would identify it as the "hiding but cloned but still existing until eventually replaced by its clone" row. More notes are in the comments in my updated jsfiddle:
Have it working a lot better now, only now the successful reload of the data is random: