That's a difficult task. I would keep a log for sure.
You said "Our parenting plan has provisions for different holidays, so in realty the amount of overnights with each parent is slightly different every year. It was stipulated that we will alternate every year which of us "gets" 183 overnights for the purpose of claiming him as a dependent."
But that must physically happen. It is allowed for you to arrange the children's schedules so that the child spends more than half the year with the father one year and more than half with the mother the next year so that you are each the custodial parent in the year you claim the child, so that you can claim full benefits. But, in practice that requires cooperation. Even then, the IRS is going to be skeptical if you force a confrontation.
My opinion (and it's only an opinion), the IRS is going to side with the legal custodial parent.
