I mostly agree with your self-answer, except for the notion that John Anderton would have been unlikely to uncover anything else.
This is not only about the murder of Anne Lively, it's also about the most fatal flaw of the whole PreCrime system, the minority reports. That's exactly what John talked to Lamar Burgess about.
At 00:34:01, John says to Lamar:
No. No, the third prevision, Agatha's prevision, wasn't there. That's
not all. I spent a few hours down there, and there are a dozen more
cases with missing previsions.
John not only shows Lamar that he's onto the Anne Lively case, he also reveals that he has spent hours down in the "halo-yard" investigating, finding out about a dozen more missing previsions. The reason these previsions are missing is because they were minority reports and thus intentionally deleted to cover up a fatal flaw in the PreCrime system.
Minority reports contradicted the other previsions, putting the accuracy of the whole system into question.
John has demonstrated that he is very interested in this. Given John's intelligent, idealistic and uncompromising nature, it's not unreasonable at all of Lamar to suspect that if he doesn't deal with this, John will continue to investigate it and perhaps even blow the whistle. Since Lamar is guilty of not only Anne Lively's murder but also of covering up the fatal and criminal flaw in the PreCrime system, it only makes sense that he does not want to take any risks with that. If I was Lamar and believed there was even a 5% chance that John would go ahead with his findings (I personally believe the chances would be much higher, by the way), it would certainly give me many sleepless nights.
Moreover, due to his central position at PreCrime, if John had blown the whistle, people would have probably believed him. As Lamar himself says at 00:35:38:
People trust you, John.
In addition, PreCrime was already at the risk of being shut down, and under investigation by a federal agent (Daniel Witwer) from the DOJ who was specifically looking for flaws (00:27:39). All John would have had to do would be to mention his findings to Witwer. That's how dangerous this situation was to Lamar. Such revelations would probably have been the final nail in the coffin for PreCrime. And indeed they were, as we saw when all these things were exposed at the end of the movie.
Actually, Lamar's past shows that he was willing to commit murder over much less. For example, Anne Lively said that she wanted her daughter (Agatha) back. At that point, Lamar could have granted her request. It would have meant the end of PreCrime, but it wouldn't have incriminated Lamar himself. And still, Lamar was willing to murder an innocent woman (Anne) just so save his brainchild, PreCrime. That's how devoted to and manic he was about PreCrime.
This time, he's in even deeper trouble. It's not only PreCrime that will be shut down if John comes forward, it's also Lamar's whole life. PreCrime will be shut down and Lamar himself will go on trial for the murder of Anne Lively. So, since he was willing to murder Anne Lively when there was much less at stake, the fact that he is willing to frame John now, when there's even more at stake is not surprising. Actually, it's completely in character for Lamar.
To summarize: It seems possible, if not likely, that John would ruin both the system Lamar loved and had devoted his life to, as well as expose Lamar himself as a murderer. Given Lamar's ruthless character, it is not at all surprising that he came up with a nefarious scheme in an attempt to prevent that. His entire life was at stake, and I think it makes total sense that he tried to deal with the contingency early on.