I think there is a way to talk to them without interfering in their personal lives. Just call them for a meeting, and then start explaining that the intermittent bickering happening between them is adversely affecting the flow of the project, and they need to be more professional. I'm sure this will change things for the better, because they will realize that their fighting in the company may jeopardize their involvement in the project, and to some extent, their jobs.
Please let me know what happens.