Dedicate one sprint to figuring out how to keep the code functioning properly mid-refactor. This can take the form of deprecated methods and classes, wrappers, adapters, and the like. Your refactoring may make the code dirtier for a short time in order to be cleaner long-term; that's OK. Right now you're saying it can't be done. I don't think that's right - think about what your process would be like if it could be done, think about what steps you can take to make it so. Then dive in.