Hi Nicholas,
Its actually not a field that is supported by export (kind of odd). Try creating a custom field and then build a smart campaign that copies the values of the Last Interesting Moment Desc into this custom field. Now you can export that.
Its also weird that you actually can't filter by it in smart lists either. But to top it all off...you can actually see its token...ha...so the above should work for you.
Just change the data value of your new custom field : {{lead.Last Interesting Moment Desc}}
You can do try it on a single lead first and see if it works and then do an automated smart campaign to have it batch everynight or trigger.
Let me know how it goes.
Cheers,
Michael