Here is what I do to diffuse arguments between me and my co-founder when we have reached the point where all the information has been presented and it's starting to get repetitious.
Agree that both options are off the table now and that a third solution must be found.
Normally if there is an argument both parties do have actual legitimate concerns that need to be addressed. The problem is that often once you get into argument mode, your ability to be creative and modify your solution to fit the other persons concerns can go out the window.
By removing both options you can go back into creative mode.
A lot of the time when doing this we actually have arrived at a much better idea than either of the two presented before that makes us both happy.
Often you end up settling on an idea that is 95% the same as what one of the people suggested in the first place, only now it also addresses the concerns of the other person too and they will now feel on board with it, which is a much better situation than if one person had "won" the argument leaving the other bitter.