Given a space $X$, construct a CW complex $L(X)$ s.t. they have the same fundamental group

Your mistake is where you say that $QR$ corresponds to the loop $a^{-1}b$. Usually it does not correspond to $a^{-1}b$ but rather to another loop in $X$ that is homotopic to $a^{-1}b$ in $X$, but not in the wedge of circles constructed in step 1 of your question. The $2$-cell $e^2_\tau$ is added to $L(X)$ precisely for the purpose of producing a corresponding homotopy in $L(X)$.