John, I have worked on more than 15 projects so far so what I am going to share is my experience.
Usually, what you are experiencing is a typical problem, and we all have faced it at one point or another. You can solve this by following a simple approach:
1. Gather information about pain points, User Journey
2. Devise a Digital Transformation/Change strategy
3. Prepare a plan to evaluate your plan/strategy
4. Be ready to Answer all concerns/question from you customer
5. Implement the A/B testing
Start asking more question from your business about How have they identified that they need to redesign? Do they have metrics to support that, this can be in the forms of analytics reports such as increases bounced rates/high exit rates from a particular page? The reason I ask is that with so much qual/quant data available to us, we can empirically identify not only that there is a bad experience, but where and why. With this information, you can start looking at ways to improve the experience. You need to point out the issue with the page or site distinctly.
Once you have Data(gathered from the client) to support your Strategy, Go back to your customer with your approach and be ready to answer all their question/pain points, How you are going to solve the problem and how you can measure the success with you AB testing strategy.
Ground up transformation are risky and generally should be avoided by companies. If the issues go beyond aesthetics or the problems are glaring and no longer can be remedied with band-aid fixes, can't be fixed by changing a component or repositioning of component/content then it is best to rip off the bandage and fix the problems. There is no point doing AB testing on terrible sites... which brings up the old saying "polishing the turd" - if your site is awful then consider reworking it from the ground up using CRO/UX techniques.
If your site/page is lacking in certain areas that need improvement, then develop data-driven test ideas and try an A/B testing tool.