Would love your feedback on a tool I am making that I think does exactly that. Still alpha, but would appreciate any feedback on if this solves your problem
https://voodoocall.com/docs/
tl;dr: configure responses to your 3rd party http dependencies, matching against header or body or url, and returning a custom response, including headers.
The best feature, I think, is that you can pass a header in that configures the response, so you can have tests that dynamically force response content/errors. That and allowing your integration tests to validate error paths and various states.