Clearly I need to read more carefully, indeed it is BAM. I also did not note that the X by E was for TO, but since partner and I both have diamonds and W had a natural NT call I didn't need to look.
My carelessness together with your note that XX did not occur to you supports my general contention that the loss of focus very often where to look when we ask where we went wrong.
I have been thinking more about the merits of 3D. The casual version of LOTT says that since we have at least nine diamonds we can bid to the 3 level, but I have never cared for that version. The original version was more in line with the name. The number of tricks we can make in diamonds, added to the number of tricks that they can take in clubs is often the same as the number of cards we hold in diamonds plus the number of cards they hold in clubs. Of course at them time that E made his double W had not yet selected clubs. However, N doubled 1C and then raised our diamonds after the 1NT. EW are showing strength and so I am not so sure N would bid 2D if he held, say, 4=3=4=2 shape. Maybe we should play N for either a fifth diamond or, more likely, a stiff club. 4=4=4=1 would make 2D seem right. Perhaps a stretch, but not unreasonable. If we give him that shape then we have nine diamonds and they have nine clubs, so LOTT says that if they can make exactly 3C then we can, likely, make exactly 3D.
Now LOTT is not right as often is its enthusiasts claim, but it seems to be about right on the average. And we have no objection when LOTT underestimates the trick total. So if LOTT is right, or if LOTT has underestimate the trick total, we might be fine with 3D. LOTT is based on double dummy play, but of course we will be brilliant.
I think I am stretching, maybe stretching quite a bit, but I have been trying to see why 3D is right over the X and this is what I have come up with.