This is the simplest and fastest way to stitch photos from Aperture with DoubleTake.
When you drag a photo from Aperture it is the the JPEG preview of your latest adjustments which gets into DoubleTake. This is fast because the previews have already been generated from the originals. However you may not wish that Aperture generates full resolution previews if your hard drive is running low on disk space, so you may adjust the size of the previews Aperture generates. Smaller than full size previews are fine for panoramas as you are making up for this by stitching several photos together.
This 14 second screen recording demonstrates Drag and Drop from Aperture
If you use TIFF DoubleTake will also save as TIFF when you use the Save to Aperture (or iPhoto) shortcut in the tool bar. When you make any changes to a RAW photo (NEF, CR2 - each camera maker has its flavour) you need to use TIFF to preserve all quality.
27 Second screen recording showing Aperture's export function (here JPG images - TIFF images are larger and it takes a bit more time)
The Service Menu in Mac OS is very flexible and you can download this service plugin for stitching with DoubleTake.
Aperture offers access to its images via a plugin interface.
This will save you the step of picking a spot to save exported images in Finder, but otherwise be identical to Export.
Update: this is among the top features I will do. Getting ready for the Mac App Store is more pressing for the moment.
Update: The Aperture Service workflow is more flexible than a plug-in will be, and perhaps it may end up being more interesting than a plug-in.