Currently unstable builds in Jenkins are marked as failed in Bitbucket and I don't see a way how to configure it differently

https://github.com/jenkinsci/bitbucket-branch-source-plugin/blob/8c2c20f4443c11798a032a9f03523c73befd6296/src/main/java/com/cloudbees/jenkins/plugins/bitbucket/BitbucketBuildStatusNotifications.java#L119

There may be reasons why an unstable build still may be considered as "successful" in Bitbucket, even so the Jenkins build is unstable - i.e. you have a threshold set which allows certain tests to be failing but still consider the Jenkins test as not failed. The plugin should respect that.
As such builds are considered failed, the code cannot be merged if the merge checks are available.

The best solution would be to have a parameter "[ ] consider UNSTABLE builds as SUCCESS" where you can define the behavior

• per team folder
• jenkins wide