This is challenging indeed and something I've had to deal with at every company I've worked for. What I've fund helps keep me and the business teams sain is to plan to launch features 14 days after the official planned released date. This makes product nervous most of the time, but most of the time they're also delayed so it all works out in the end.
Goes back to the shared goals - which at a high level, are hard to argue with - revenue, cost savings, customer success, etc. Once you get that common agreement, then it's about the strategy / the "how" to get there. If there are disagreements here, I would start with trying to understand why and seeing it from both of their vantage points. Then trying to see if you can get them 1:1 to understand the other point of view or better yet, get them to talk to each other. Ultimately though if all that doesn't work, you may need to get a tie breaker that's someone else and who they will listen to.
I must admit that design disagreements have been rare in my experience. The best way to ensure alignment is to really think of your design partner as a true partner (vs. just a service role), bring them in early and upfront, provide them with context (e.g. maybe even have them be part of planning sessions), and take a collaborative vs. directive approach.
While there may always be one-off disagreements on individual efforts, overall a collaborative approach should lead to a much better working relationship and end result since you’ll now be aligned on goals and desired outcomes.