Is the Nash product really maximised ex post?
The Nash bargaining solution DOES maximize the Nash product. You have to separate the playing of the game from the bargaining problem. If the players negotiate a binding agreement they will realize that their maximal total payoff from playing the game is $3$. This can be achieved by playing $(T,L)$ or $(T,C)$, or any mixture of those two profiles.

The bargaining solution just asks: Given a total payoff of $3$, how will the players divide this payoff among them, given a threat point of $(0,0)$? So the efficient frontier is (a part of) the line $u_1+u_2=3$, and by symmetry (or by maximizing the Nash product $u_1(3-u_1)$) you find $u_1=u_2=3/2$. The only remaining question is how to play the game (under a binding agreement) in such a way that these are the expected payoffs. But you already solved this part.