I don't think that that moment was out of character for the Joker. I thought that he wanted Lex to kill him, rather than just let him be tied up, because, as Joker mentioned himself in part one of that episode, he knows how the Bat thinks. He would know that, if left alive for that long, Batman would escape, which would ruin their plan, as well as a chance to finally kill him.
I understand the idea that Batman gives Joker a reason to live, but I think of it more like Batman gives Joker a reason for his crimes. From nearly all of the episodes involving the Joker and Batman fighting, it seems like Joker acts out his crimes as games. A key part of those games involves intense struggles between him and Batman. Without that conflict, Joker could easily get away with most anything. The Joker himself mentioned that in "The Man Who Killed Batman" when he said: "Without Batman, crime has no punch line."I always noticed that Batman was the only one who could really challenge the Joker, while everyone else in Gotham pretty much were struggling to figure out his plans.
The act of killing Batman almost feels like an attempt to win the game that Joker always plays with him. If he doesn't feel like he did it or was a part of it, he acts either depressed, as he did in the already mentioned BTAS episode, or becomes extremely mad, like he did in "Mad Love." If he was able to convince Lex to take him out right then and there, not only would he be a part of his death, but he would be able to stand right there and watch it happen. Of course, Batman wouldn't let that happen because he's Batman.
As for the whole unmasking thing, I've always thought that was kind of strange. Batman has been captured before, but rarely do the criminals try to unmask him. The only time I could remember that happening was at the end of "Trial." But that was because Joker wanted to save that for the end of the trial anyway. I guess that for all of the people who captured him that his death was more important than finding out who he is. They could easily find out his identity after taking off his mask, if they didn't know what Bruce Wayne looked like, but they would probably get more street credit and fame for killing him as Batman, not as Bruce Wayne.