Problem being, if you challenge another system of morality, all you have to challenge it with is that your own system contradicts it. Without an objective place to stand (or at least the belief that your system is objectively correct), it's your word against theirs.
You are arguing that morality is not relative merely because you would not be able to have some objective, logical reason with which to convince others that the morality you perceive is correct. Just because it would be handy does not mean that it exists.
It is not really a matter of your word against theirs. Assuming you know that, since morality is relative, when you say that what they are doing is wrong, you mean that it is wrong
according to you, and, when they say that it is wrong
according to them. Now, assuming again that there is no objective morality, you are under no obligation to accept what they are doing, and you are under no obligation to adhere to any kind of moral code in your efforts to make them conform to your view of right and wrong, so, obviously, if trying to convince someone that what they are doing is wrong, you would not mention the fact that morality is relative, and you might appeal to their empathy or whatever, but, if they do not relent, you would be perfectly justified in using force to stop them.
“We would have no way to win an argument with someone that has conflicting views,” really does not mean that that there is such a way. That is an appeal to consequences.
One of my college profs pointed out the problem with moral relativism this way:
"I believe that there is a leprechaun in the corner who talks to me."
Everyone agreed that this was OK.
"The leprechaun just told me to shoot you."
“Moral relativism is wrong, because it means I get to SHOOT YOU whenever I feel like it.”
Well, all right, the Leprachaun told you to shoot me. I do not like it, but that does not mean that there is some objective reason that you should not do it. Yes, you do have objective leave to shoot me. What of it? I also have no objective obligation to like it or tolerate it.
Moral relativism is easy until the rubber hits the road.
I think it easier than you think it is, but ease does not imply correctness anyway.
If you are going to argue for some objective moral code, you will need to provide the observations and logic that would lead to those morals. You can not say that something exists in an objective fashion without evidence or reasoning.