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
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.