From the start:
1. Decide who's going to take point on fixing this (one single person). This person has the final say on how it will be done.
2. Get input from all parties to extract as much information as possible, with the understanding that a lot of it will involve conjecture and opinion, and that the pointman will decide what to do in the end. This changes the relationship from rival to lobbyist (an important psychological distinction).
3. The pointman thanks the other person for their input, and then implements their best judgment based on the information at hand.
4. If the solution fails, reconvene, assess whatever new information came out of the attempted solution (without judgment, blame, or I-told-you-so), and then decide on the next plan of attack (Is there already another avenue to try? Do you need to do more research and reconvene again? Do you want to switch who takes point?)
5. Collaborate to write a blog post about what you learned. If two professionals have such divergent opinions on the solution, it's likely a difficult problem with many gotchas that others will appreciate reading about.
Difference of professional opinion is an empowering thing, offering learning and even potentially the discovery of new knowledge; the trick is to channel it skillfully to a constructive end.