You can use multiple remote repositories with git. But you'll have to push separately into 2 of your remotes.
cd project
$ git remote add github https://github.com/your_user/your_repo.git
$ git remote add bitbucket https://[email protected]/your_user/your_repo.git
Now in order to push changes to corresponding branch on github or bitbucket you can do this:
$ git push github master
$ git push bitbucket master
Same rule applies to pulling: you need to specify which remote you want to pull from:
$ git pull github master
$ git pull bitbucket master
If your project currently points to github, you can rename your current remote repository to github:
$ git remote rename origin github
Finally, to see which remote will be fetched from:
$ git remote -v show