Is there a way to implement a commit message policy on BitBucket Cloud? From my understanding, the webhooks only work as commit notifications and can not intervene with the commits (to deny the commit if the commit message does not follow the set regex). For pre-commit hooks you would need BitBucket Server, right?
If that's the case, what about checking commit messages on a Jenkins build to fail the build if it contains a certain message that does not match the regex? Is that a viable option?
Another option, is there a local git config file where I could set the rules locally (even if they could be bypassed)?
I think you're confusing pre-commit hooks (which determine whether or not a commit can be made) with pre-receive hooks (which determine whether or not a specific remote will accept pushes). Bitbucket Cloud does not currently support custom pre-receive hooks, but Bitbucket Server does. Pre-commit hooks would need to be installed on every system where users make commits; those are not currently replicated, so you'd need everyone to manage them manually.
It sounds like Jenkins tests would be your best bet - check the commit message for the desired regex, and fail the test if the regex isn't there. That won't prevent people from making bad commits in the first place, but it will keep those bad commits out of the final product.