The person to call out a lie in a debate is the opponent.
This.
Otherwise, moderators set themselves up as being part of the debate, becoming a visible opponent.
I'd love to test run this for OP or anyone who thinks it would work out swimmingly for such a moderator. In essence, who will determine (ultimately) what the actual truth is? The moderator? What if they are wrong, and that is shown post debate? Then that would most definitely show the moderator injected themselves into the debate (becoming a de facto debater) and if errant at any point on 'presenting and discussing facts' were very poor at both their job (moderator) and debating.
Not to mention the fact that we have plenty of people already spinning things post debate, that it really isn't necessary to get to the truth during the debate, as what really matters is 'given their assumed knowledge on the issue, what is their proposed policy.' That's what we need to know. If post debate it shows their assumed knowledge is incorrect, we (the press-American people) can then ask them in light of that information, would their policy be different.
I fully expect both participants in these debates will stretch the truth, and/or spin it to match what their proposed policies are aiming for. If there is a gross lie being perpetuated (i.e. all Mexicans alive today are rapists), I would expect the opponent to call that out. If the moderator did, I would expect it to be in form of a question and let the person respond.