$u\in{\operatorname{dom}}(\partial_{t,\rho})$ we have that $$(1+\varepsilon\partial_{t,\rho})^{-1}u-u=-\varepsilon(1+\varepsilon\partial_{t,\rho})^{-1}\partial_{t,\rho}u\to0\quad(\varepsilon\to0+)$$ the assertion follows.\
We now prove that ${\operatorname{dom}}(\partial_{t,\rho})$ is a core for $\partial_{t,\rho}\mathcal{M}$. More precisely, we show that for $u\in{\operatorname{dom}}(\partial_{t,\rho}\mathcal{M})$ and $u_{\varepsilon}\coloneqq(1+\varepsilon\partial_{t,\rho})^{-1}u$ with $\varepsilon>0$ we have that $$\partial_{t,\rho}\mathcal{M}u_{\varepsilon}\to\partial_{t,\rho}\mathcal{M}u\quad(\varepsilon\to0+).$$ Indeed, we have that $$\begin{aligned}
\partial_{t,\rho}\mathcal{M}u_{\varepsilon} & =\partial_{t,\rho}\mathcal{M}(1+\varepsilon\partial_{t,\rho})^{-1}u\\
 & =(1+\varepsilon\partial_{t,\rho})^{-1}\partial_{t,\rho}\mathcal{M}u+\varepsilon\partial_{t,\rho}(1+\varepsilon\partial_{t,\rho})^{-1}\mathcal{M}'(1+\varepsilon\partial_{t,\rho})^{-1}u\\
 & =(1+\varepsilon\partial_{t,\rho})^{-1}\partial_{t,\rho}\mathcal{M}u+\mathcal{M}'(1+\varepsilon\partial_{t,\rho})^{-1}u-(1+\varepsilon\partial_{t,\rho})^{-1}\mathcal{M}'(1+\varepsilon\partial_{t,\rho})^{-1}u\\
 & \to\partial_{t,\rho}\mathcal{M}u\quad(\varepsilon\to0+).\end{aligned}$$

Hence, it suffices to approximate elements $u\in{\operatorname{dom}}(\partial_{t,\rho})$ by a sequence $(\varphi_{n})_{n}$ in $C_{c}^{\infty}({\mathbb{R}};H)$ such that $\varphi_{n}\to u$ and $\partial_{t,\rho}\mathcal{M}\varphi_{n}\to\partial_{t,\rho}\mathcal{M}u$ in $L_{2,\rho}({\mathbb{R}};H).$ For doing so, we choose a sequence $(\varphi_{n})_{n}$ in $C_{c}^{\infty}({\mathbb{R}};H)$ such that $\varphi_{n}\to u$ in $H_{\rho}^{1}({\mathbb{R}};H)$. Then, in particular, $\varphi_{n}\to u$ in $L_{2,\rho}({\mathbb{R}};H)$ and $$\partial_{t,\rho}\mathcal{M}\varphi_{n}=\mathcal{M}\partial_{t,\rho}\varphi_{n}+\mathcal{M}'\varphi_{n}\to\mathcal{M}\partial_{t,\rho}u+\mathcal{M}'u=\partial_{t,\rho}\mathcal{M}u.$$ We now prove the maximal monotonicity of $\partial_{t,\rho}\mathcal{M}+\mathcal{N}-c$. As $C_{c}^{\infty}({\mathbb{R}};H)$ is a core for this operator, the monotonicity follows by (b). We claim that $C_{c}^{\infty}({\mathbb{R}};H)$ is also a core for $\left(\partial_{t,\rho}\mathcal{M}+\mathcal{N}\right)^{\ast}$. Assuming that this is true, it follows from (b) that $\left(\partial_{t,\rho}\mathcal{M}+\mathcal{N}\right)^{\ast}$ is one-to-one and hence, $\partial_{t,\rho}\mathcal{M}+\mathcal{N}$ has dense range. Since $\partial_{t,\rho}\mathcal{M}+\mathcal{N}$ is also closed and its inverse is bounded by what we have shown before, $\partial_{t,\rho}\mathcal{M}+\mathcal{N}$ is indeed onto and thus, $\partial_{t,\rho}\mathcal{M}+\mathcal{N}-c$ is maximal monotone. Thus, we are left to show that $C_{c}^{\infty}({\mathbb{R}};H)$ is a core for $\left(\partial_{t,\rho}\mathcal{M}+\mathcal{N}\right)^{\ast}$. For showing this, we observe that $$\left(\partial_{t,\rho}\mathcal{M}+\mathcal{N}\right)^{\ast}=(\partial_{t,\rho}\mathcal{M})^{\ast}+\mathcal{N}^{\ast}=(\mathcal{M}\partial_{t,\rho}+\mathcal{M}')^{\ast}+\mathcal{N}^{\ast}=\partial_{t,\rho}^{\ast}\mathcal{M}^{\ast}+(\mathcal{M}')^{\ast}+\mathcal{N}^{\ast},$$ where we have used that $C_{c}^{\infty}({\mathbb{R}};H)$ is a core for $\partial_{t,\rho}\mathcal{M}.$ 