You could simply add a second remote:
git remote add bitbucket /url/to/am/empty/bitbucket/repo
and push everything to bitbucket:
git push --mirror bitbucket
You can actually pull from or push to multiple remotes from your local repo.
Update 2020:
As noted below in Rahulmohan Kolakandy’s answer, if you are talking about an on-premise BitBucket server (as opposed to bitbucket.org), then you can take advantage of BitBucket Server Smart Mirroring.
As commented by V-Q-A NGUYEN:
BitBucket Server Smart Mirroring (introduced originally in 2016, and Oct. 2017 for BitBucket Server)
is only available for customers with an active Bitbucket Data Center license