There is no way to correctly implement a service account on Bitbucket Cloud.
Formerly, SSH keys on team/organization account could be used for this purpose, but for some time now, the server is emitting this sort of warnings
remote: Warning! remote: You are currently connecting with your team account.remote: This is no longer supported, so please connect using your user account.
The recommended way of providing automation seems to be using SSH Access keys, but these only allow Read-Only access, no write access.
Asking on the Atlassian Community, the only answer I got was to use my personal account as a service account, which is unsatisfactory.
Please provide SSH Access keys with read-write access on Bitbucket Cloud.