JenkinsFile to get git branch name
If you check the Git plugin documentation, you can see that there is a GIT_BRANCH environment variable that you can use in your pipeline.

GIT_BRANCH - Name of the remote repository (defaults to origin),
             followed by name of the branch currently being used,
             e.g. "origin/master" or "origin/foo"


So, in your pipeline:

stage('Push to Bitbucket') {
    steps {
        withCredentials([usernamePassword(credentialsId: '8e2ca0a6-2dc5-4f89-b76a-03076d6b0843',
                                          passwordVariable: 'GIT_PASSWORD',
                                          usernameVariable: 'GIT_USERNAME')]) {
            sh '''
            # Content omitted
            git checkout ${GIT_BRANCH}
            # Content omitted
            '''
        }
    }
}