No, The IRS doesn't care about the "TN legal system". They go by physical custody and there are no half days allowed. Yes, they expect you to count the number of nights the sleeps at each home. And yes you may arrange your schedule so they you can meet the technicality on alternating years. Remember, you have to coordinate with the other parent. You can't be claiming HoH if she thinks she gets to claim HoH.
You are correct, it's complicated. There is no simple answer to "will it hurt me financially to change our plan" . Probably yes. But the only way to know for sure is to do test returns. This is more easily done with desktop TurboTax (TT) rather than online. Or you can use this tool: <a rel="nofollow" target="_blank" href="
https://turbotax.intuit.com/tax-tools/calculators/taxcaster/?s=1">https://turbotax.intuit.com/tax-to...>.
Be careful about insisting that you have joint custody. In the rare case (could probably only happen in a leap year like 2016), where the time that each parent has the child is exactly equal, then the parent with the higher income (AGI) is the custodial parent, for the purpose of determining who has first priority on claiming the child as a dependent. But then neither parent can claim a Qualifying Child dependent, for some tax benefits because neither parent had the child the required MORE than half the year. (no earned income credit, based on that child, and the child would not qualify the parent for Head of Household filing status).
