Can you setup automated integration/end-to-end tests and find a way to feed that back into your AI agents before a human looks at it? Either via an MCP server or just a comment on the pull request if the AI has access to PR comments. Not only is your lack of an integration testing pipeline slowing you down, it's also slowing your AI agents down.
"AFAICT, there’s no service that lets me"... Just make that service!