Do you know a reference for the following property ?
Let $X$ be a Banach space and $D$ be a dense subset of $X$. Let $(T_\alpha)$ a bounded net of bounded operators on $X$ such that the net $(T_\alpha(x))$ converges in $X$ for any $x \in D$. Then there exists a bounded operator $T$ on $X$ such that $(T_\alpha)$ converges pointwise on $X$ to $T$.
Rem: I know references if the net is a sequence of operators.
You should check if the proof given in your references works if you substitute "sequences" by "nets". If that's the case I would just cite the "sequences"-reference and give a remark, that the same proof also works for "nets". Anyway, a direct proof isn't too much work:
Note that the crucial point is that $(T_\alpha)$ is a bounded net of operators.
Furthermore, due to the linearity of the $T_\alpha$, we can assume that we have convergence not only on the subset $D$ but also on $\operatorname{span} D$, since for $x = \sum_{i = 1}^n \lambda_i x_i$, with $x_i \in D$, we have $$ \lim_\alpha T_\alpha x = \lim_\alpha \sum_{i = 1}^n \alpha_i T_\alpha x_i = \sum_{i = 1}^n \alpha_i \lim_\alpha T_\alpha x_i. $$
Then, we define a mapping $T$ on $\operatorname{span} D$ by setting $$ T x := \lim_\alpha T_\alpha x. $$ for all $x$ in $\operatorname{span} D$. This mapping is clearly linear, since $T_\alpha$ is linear for all $\alpha$ and converges pointwise to $T$ for all $x \in \operatorname{span} D$.
Now, $T$ should be bounded on $\operatorname{span} D$, since for all $x \in \operatorname{span} D$ you have $$ \|T x\| = \|\lim_\alpha T_\alpha x\| = \lim_\alpha \|T_\alpha x\| \leq \lim_\alpha \|T_\alpha\| \|x\| \leq (\sup_\alpha \|T_\alpha\|) \|x\|. $$
But then, $T$ can be continuously extended to all of $X$, since $D$ and accordingly $\operatorname{span} D$ are dense subsets of $X$.