I agree with the sense that pervades all of the Stack Exchange discussions of Memento. There is much here left to inference.
But here is my inference, at least at the moment.
As <PERSON> was confronting <PERSON>, he wondered how he would remember that he had done it. He worried (with excellent accuracy, it seems) that he might mistakenly continue to pursue new suspects after killing the real offender. So <PERSON> decided to leave himself a reminder that he had killed someone by changing into his victim’s clothes.
But then <PERSON> said "<PERSON>," and <PERSON> showed up and confirmed that everything was different from what it had seemed. So <PERSON> decided to commemorate not that he had killed the true offender, but (falsely) that <PERSON> was the true offender; he recorded <PERSON>'s license plate.
<PERSON> drove to a tattoo parlor, and began to get a new tattoo. <PERSON> followed him there, and into the actual workspace, to persuade <PERSON> to change out of <PERSON>'s clothes. 861
<PERSON> might want <PERSON> to change because dressing as <PERSON> might get <PERSON> killed. But for this purpose, why did <PERSON> actually enter the room in which <PERSON> was receiving the tattoo? Why didn't he wait until <PERSON> emerged? Perhaps <PERSON> was worried that <PERSON> would commemorate, with another tattoo, that he had succeeded in killing "<PERSON>" To prevent this he would have to interrupt the tattoo already in progress. This is ironic, of course; <PERSON> is there to commemorate not that he had killed the offender, but that <PERSON>'s license plate was that of the offender. And eventually, <PERSON> does in fact tattoo himself with the message that he had succeeded in killing "<PERSON>" only after killing <PERSON>.
When <PERSON> left <PERSON>'s house, he found <PERSON> waiting in the Jaguar. <PERSON> asked <PERSON> how he came to be wearing a designer suit and driving a Jaguar; had he done these things while working for the insurance company? Of course the answer to this question should be "no"; the new clothes and car should suggest to <PERSON> that there's something he's misunderstanding about his new situation. This conversation thus becomes part of a major theme of the film, that to a very large extent we all fool ourselves all the time, failing to see evidence that leads in a direction that we do not want to be led. (Another example of this obtuseness is the scene in which <PERSON> appears before <PERSON> with new bruises on her face, and <PERSON> flexes his hand, noticing that he seems just to have punched someone; but he fails somehow to link these things.) 861