Since this is self-study question, I will not provide any solution but rather jot down the general approach to attack the problem.
It has been shown $(T_1, T_2):= \left(Y_{(1)}, \sum Y_i\right)\equiv \left(Y_{(1)}, \sum \left(Y_i-Y_{(1)}\right)\right)$ are jointly sufficient for $(\mu, \lambda).$
What should be the approach to show $(T_1, T_2)$ is jointly complete?
$\bullet$ Stick to the definition (unlike what OP attempted): Show that $$\mathbb E_{\mu,\lambda}[f(T_1, T_2)] = 0 \implies f\equiv 0 ~~\textrm{a.s.}\tag 1\label 1$$
$\bullet$ Show that $T_1, ~T_2$ are independently distributed. Find out their respective distributions.
$\bullet$ For fixed $\lambda,$ apply Fubini's theorem in $\eqref 1$ to get $$\int\underbrace{\left[\int f(t_1, t_2)f_{T_2}(t_2)~\mathrm dt_2\right]}_{=~\mathbb E_\lambda[f(t_1, T_2)]}f_{T_1}(t_1)~\mathrm dt_1 = 0\tag 2\label 2$$ for all $\mu.$
$\bullet$ Conclude $\mathbb E_\lambda f(t_1, T_2) = 0~~\textrm{a.s.}$ (How?)
$\bullet$ Finally show the above conclusion is true for all $\lambda.$ The rest follows. (How?)
$\rm NB.$ The approach is based on the assumption that $\mu\in \mathbb R, ~\lambda \in \mathbb R_{+}.$ Can OP modify (Is it necessary or not? Check.) the argument for $\mu\in\mathbb R_{+}?$
Reference:
$\rm [I]$ Theory of Point Estimation, E. L. Lehmann, George Casella, Springer-Verlag, $1998,$ sec. $1.6,$ p. $43.$