If he does - or decides later that he wants to - then he does have the trump card here as the actual, you know, star player. He can make it clear that LaVar does not speak for him, sever business ties with his father (and if need be / dragged to those lengths, personal ties too), and even ask that his father not be allowed to attend games if he's going to continue to comment to the media. People can only talk about the interpersonal drama just so long when the actual famous side in it refuses to give them further fodder.
Hell, he could ask the team to threaten to refuse credentials to anyone who interviews his father about the lakers. They can interview the crazy hobo down in Skid Row about the lakers too, but don't expect to be taken seriously as a reporter afterwards, right? Point is, Lonzo can put down a firm foot if he wants to. His entire life seems to have been prelude to him being emotionally unable to do so, but he's 19, he'll start feeling like an adult sooner or later.