I was able to solve this using this suggestion:
- Remove the SSH key from the repo. (Click on repo name > Settings > Access Keys)
- Add SSH key to Account settings SSH keys. (Click on your avatar > Bitbucket Settings > SSH Keys)
So, adding the keys under the repo only gives you a read-only access. For read and write access, you need to add your keys under your account.
To be fair, Bitbucket warns about the read-only access when adding the key to the repo:
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