At a quick glance, it sounds like you just need to turn on the use of "smart previews". This will allow you to edit and export when the external hard drive is disconnected.
Which version of Lightroom are you using?
You could create Smart Previews of all your Pictures that will allow to edit them without the External drive connected. See How to use Smart Previews to view and edit photos in Photoshop Lightroom
Ah! I'll have to give that a try. I read Adobe's help page on smart previews (How to use Smart Previews to view and edit photos in Photoshop Lightroom), but it doesn't even mention exporting, so I assumed that it wasn't possible.
Before you go crazy creating lots of smart previews, do a test to make sure exporting works without the original files. I may have gotten that wrong.
Smart Previews are really smaller 'DNG' versions of your full-size files.
they have a 'Long edge' dimension of 2540 pixels, and are quite suitable for prints up to 8x10in.
Not only can you use Smart Previews to Develop images when they’re off-line, you are also able to use them in the Publish Services panel (Facebook or Flickr) and Export them as JPEG. You can use them to layout a book (you will not be able to print the book until the original images are on-line as the quality of the Smart Previews will not be high enough to print), and create and output a video slideshow (although again, they can not be used to output a slideshow to PDF because of quality concerns) and create and export web galleries. The best rule of thumb; for the highest quality, you’ll most likely want your original files when outputting files.