Any private gitbucket/bitbucket integration on the card, like most CI SaaS do, gets an oauth2 token for the desire repositories, setups the notification webhooks, deploys on push (of nominated branch)?
We're looking into building tools like this, but also note that you can whip up something yourself that pushes via the Workers API. Maybe you can even implement it as a Worker... :)
Lack of tooling integration is one of the most challenging things about using services like yours in production. "Whipping something up" is not something I want my eng team doing. Having built a system where customers want a similar integration, I understand how difficult it is to provide!