Also no spotlight or fawning media in that scenario, and possible re-election difficulties...
That's exactly it. If he formally became a Democrat, then he would just be another Democrat. If he stays in the Republican Party, then he is a traitor and a turncoat. He gets special points for that with the media.
Also, he didn't want to disrupt his reelection. The entire voting system of AZ has been rigged to make sure he gets reelected every time, even if he happens to be dead at the time of the vote. With the way the system is set up, death should not make a difference in his reelection. Switching party would have meant taking a risk. And there was no need for it. He votes like a Democrat anyway, he 'caucuses' with the Democrats, so as long as he gets elected, party label makes no difference in terms of what he will do in office, as we have clearly seen.