You can prevent the applications on your external drive from appearing in the Open With… menu by preventing Spotlight from indexing the drive.
To do this navigate to System Preferences > Spotlight. Select the Privacy tab and add your external drive to the list. Mutineer in the screenshot below is my SuperDuper clone drive.
My best recollection is that this was all I did to solve the problem, but if the problem persists, two additional steps may be necessary:
Make Launch Services reload the list of available apps with the command /System/Library/Frameworks/CoreServices.framework/Versions/A/Frameworks/LaunchServices.framework/Versions/A/Support/lsregister -kill -r -all local,system,user. As apps on the second drive are not available via Spotlight anymore, the new database will only contain those on your main drive.
Restart Finder to rebuild its Open With context menu (Option-right click on its Dock icon to get the restart option, or do killall Finder in Terminal).
Credit to @kopischke for providing this concise answer on SuperUser.
If you want to avoid the terminal incantations, you can use Onyx to rebuild the spotlight index and launch services database.