Look mom, without hands...
First, change this in your .git/config:old url:
url = https://[email protected]/hispamedios/PROJECT.gitnew one:
url = ssh://USER.bitbucket.org/hispamedios/PROJECT.git
Edit your .ssh/config, in your local directory, and add this:
cat .ssh/config Host YOURUSER.bitbucket.org HostName bitbucket.org User git IdentityFile ~/.ssh/bitbucket
The last line is important, so you are indicating your identity file. Now create this file:
ssh-keygen -f .ssh/bitbucket -C "YOURUSER"
And now test it. Go to your local repository, do smart changes in your code and commit them or simply do a "git pull". It should not ask you for password anymore.