?iTunes library on desktop and laptop

Since your media are on the external drive, that's where they will be when you travel. No drive, no media. If you travel extensively you could consider getting iTunes Match but that's another topic.

Copying your iTunes folder from the iMac to the MBP will in essence put the file listing and other information in your primary library on the MBP but you will of course have to still use the same external drive.

Be careful about ending up with two different libraries on the two computers of you will encounter problems syncing because a device can only sync to one library at a time. You can add media from a non primary library but things start getting messy and confusing.

in its entirety on the external drive

the drive containing the iTunes media folder connected.

Inconsistency in your statement. An entire iTunes library is more than just media. What do you have on the external? Is there a whole iTunes folder there with a library.itl, artwork folder, other files and folder, or is it just media?

What are the iTunes library files? - http://support.apple.com/kb/HT1660

More on iTunes library files and what they do - http://en.wikipedia.org/wiki/ITunes#Media_management

What are all those iTunes files? - http://www.macworld.com/article/139974/2009/04/itunes_files.html

Where are my iTunes files located? - http://support.apple.com/kb/ht1391

If you want the external drive to work easily and seamlessly with both computers as far as iTunes is concerned, you need to have the entire media collection and support files (= library; a community library isn't just a pile of books in an empty lot somewhere) on the external. Then all you have to do is make sure the external is connected before starting iTunes. The first time you use it, or if you forget to attach before starting iTunes, start iTunes with the option key held down and guide it to the library.itl file on the external drive.

Use the option key startup as instructed as above and any computer with iTunes can use the library.