From Experience
I just went through this one with my wife's case. In our case her AP had been approved, just forwarded on to our old address. We went to the BCIS office in Portland OR, to try to get an emergency AP because she was due to travel in about 3 weeks. Their position was that because our case was with the NSC they had no jurisidition in the case. They also did not see that this was an emergency. My guess is that they take a very literal interpretation of emergency.
Our next step was to contact the offices of both our state Senators (they very easy to deal with and very helpful). The recomendation that came out of that and our discussions with the lawyers was that we should apply to the NSC again and request an expedited approval of the case. This involves another application and fee payment. This was on 2/28 and she was due to travel on 3/12.
The strong chorus of opinion we got from our lawyer, senators officer and from this site was that she should not travel without an approved AP document in hand.
As it turned out, her original AP document got returned to the NSC and they mailed it back on 3/7 and everything worked out in the end. The second expedited AP document did not get approved until 4/1.
Now, she was not asked to show the documents when she left the country, so did she need to be carry the documents with her? This depends on your risk tollerance.
My recomendations would be to try at your BCIS office, they may help (especially if you have a real emergency), apply again to NSC, involve your Senators office and ask them to expedite. This process is too long and hard to risk screwing it up over a holiday.
Good Luck