According to this issue FullCalendar should support the Libraries API module - http://drupal.org/node/980886
Comment #7 is especially relevant I believe.
This means you should be able to use Libraries http://drupal.org/project/libraries to handle getting your customized version of the library to be used instead of the bundled one.
Ie. Make sure it is named the same as the original and stick it in sites/all/libraries
When you update a module what ever changes you have done would automatically be over written. if you have just made changes to js then it would be better to add that JS to your theme rather than the module itself.