As far as I know when exporting an addon, only the master template and phrase will be exported and customized template/phrase will not.
Is their any way I can also export the customized template/phrase?
Those would be exported separately from the addon by exporting your custom style and language separately. But this would export your custom style and language in its entirety, not just customized templates from the addon. You can either over-export and then revert unwanted custom stuff after importing, or you can manually copy and paste specific templates through the Admin CP of both forums.
It wouldn't be that hard to write a quick script to handle this:
firstly import the addon into an array with each line an item. Do a preg_match and search for the phrase titles. Next import an export your customised languages, then spit out any line that has a phrase title that matches the first part. Wrap it the correct XML tag and give to your friend.