Yes, he sued her in 2006. And she fled her legal obligation to provide Joe with custody of Tylee more than once. Joe was entitled to custody (and ordered to pay child support) in Texas --and then she moved to Arizona. And he was entitled to custody in Arizona (and ordered to pay child support) when she moved to Hawaii.
Something in Lori's narrative and/or the retelling of the timeline doesn't line up. The idea that Lori was (ever) "ordered to pay child support" could only make sense if Joe (ever) had more custody than she did.
And, in fact, the opposite was true, because she withheld Tylee from him. And that's why I surmise that her description of this as an "order to pay child support" is actually "an order to repay Joe for some of his lost time with Tylee."