So far, from $L = P \dot{X} + \frac{1}{2}\lambda (P^2 + X'^2) + \rho P X'$ and $\tau' = \tau - \epsilon(\sigma), \sigma' = \sigma$ and $\delta X^{\mu} = - \epsilon \dot{X}^{\mu} $ and $0 = \frac{\delta L}{\delta P_{\mu}} = \dot{X}^{\mu} + \lambda P^{\mu} + \rho X'$ we have
$$\delta X^{\mu} = \epsilon (\lambda P^{\mu} + \rho X') = \epsilon \lambda P^{\mu} + \epsilon \rho X' \neq \epsilon P^{\mu} + \eta X'^{\mu}.$$
Similarly from $L = - \dot{P} X + \frac{1}{2}\lambda (P^2 + X'^2) + \rho P X'$ and $\delta P^{\mu} = - \epsilon \dot{P}^{\mu} $ and $0 = \frac{\delta L}{\delta X_{\mu}} - \pa…
