Since you don't have authority, you need to go to someone who does. These sorts of issues (how do we do X in both parts of the system so it's consistent and maintainable) typically fall to an architect, not a project manager or leader of people. If you have an architect, go and ask what to do. The architect will be able to make a good choice and enforce it.
Assuming that you don't have an architect on your team, you need to go to the people manager and explain your situation. Do not tell your "people manager" your own diagnosis of the personality types of your coworkers. Simply say that in the past, they've each implemented things differently, and they rarely agree. Explain why you feel it's important they agree in this case. And then ask your manager for something: probably in this case, it's permission to be the architect on this issue and choose a solution they both must implement. Your "people manager" can then direct both of them to understand the situation as it applies to their system, and design an approach to be delivered to you. There can then be a meeting between you, both of them, and the "people manager" in which a choice is made and the "people manager" directs them to implement it.
If you do this well, you may end up the architect. If you let it bog down in pointless arguing with each of them implementing their own incompatible thing, with the result being a permanent pain point and impedance mismatch in the project, it could be a blot on your record at the firm. So make sure this really is your battle to fight before you wade in to force them to be consistent and play nice.
A third option is to go to your manager and use this situation as an example of why you need an architect. There are even freelance architects available who will come in and "whip things into shape" for a while, then be available for quick questions and problem-settling later as required. It's something I do from time to time and I've seen the enormous difference it can make. The majority of managers will reject this option because architects are expensive, but perhaps yours will be willing to consider it.