I do not think that this is the usual definition of [Banach limit](https://en.wikipedia.org/wiki/Banach_limit). (What I know under this name is functional on $l_\infty$which is shift-invariant, see the linked Wikipedia article.)

Your conditions can be rewritten as
$$\liminf_{c\in D} f(c) \le \ell(f) \limsup_{c\in D} f(c).$$

So you want a functional which is between $\liminf$ and $\limsup$ (and therefore extends the usual limit of a net) and is multiplicative.

You can simply take any [ultrafilter](https://en.wikipedia.org/wiki/Ultrafilter) $\mathcal U$ which contains all tail sets of the directed set $D$. (I.e., for any $d\in D$ you have $d\uparrow=\{c\in D; c\ge D\}\in\mathcal U$.) And then define $\ell$ using limit along this utlrafilter as:
$$\ell(f) = \operatorname{{\mathcal U}-\lim} f(c).$$
This functional has the properties you want. (Boundedness of $f$ guarantees that the $\mathcal U$-limit exists.)

The same construction is mentioned in the answer to: [What is a generalized limit?](https://mathoverflow.net/q/242307) 

In case it helps to find some references for $\mathcal U$-limit (limit along an ultrafilter or, more generally, limit along a filter or a filter base), I will mention my answers to these questions: [Where has this common generalization of nets and filters been written down?](https://math.stackexchange.com/q/1568548) and [Basic facts about ultrafilters and convergence of a sequence along an ultrafilter](https://math.stackexchange.com/q/51476).