A lot of it is standard NBA player monkeyshines with cheating, hos, drinking, eating shitty food, partying in both the US and EU. Imagine a white and mildly less injury prone Zion Williamson. The biggest rumor was that he was hooking up with Serbian pop singer named Breskvica throughout the second pregnancy, though this might just be the Serbian internet on their bullshit as always -- every EU internet discussion of Luka will inevitably be shat up by frothing Serbian nationalists claiming him as a true pure blooded Serb only interested in other Serbs etc.
The ex Anamaria is pretty normal, lowkey and family-oriented as far as NBA WAGs go. From a wealthy family with supportive parents. She found out she was pregnant at around the same time as the trade and reeeeeeeally didn't want to go to LA where she knew Luka would have far more access to his "issues" than he did in Dallas. They more or less separated after that, she never lived in LA with him. He's apparently a bit of a manchild and she doesn't trust that he can provide a safe household for the little girls on his own and is only interested in co-parenting in Slovenia (where he lives during the off-season anyway).
In the bigger picture, Anamaria and Luka's mother don't get along and Luka's own immediate family situation is a mess (parents only communicate through him, his mother and her boyfriend spend a ton of his money, his shitheel deadbeat Serbian father only became interested in him when his basketball prowess became apparent, etc.)
He's worth hundreds of millions of dollars and this is a high profile international custody case, he would be absolutely DSP tier retarded to not get the best of the best.