I agree with part of that... I would put a stout in it, the longer you leave it the more oak flavors you will get, it will evaporate some but not much in the amount of time we are talking. I would leave it in for 10-14 days and use that as a secondary fermenter. I assume this has recently had rye whiskey in it if so you should be good... if not I would put some in and slosh it around to sanitize it, I would also leave it in there to add some extra flavor to the beer. (not much just enough to get the surface of the inside wet) maybe 1 cup.
I dont understand the DME to top it off? I would just go with what you get, technically it will evaporate out of a carboy also.