I do not think that this is the usual definition of Banach limit. (What I know under this name is linear functional on $l_\infty$ which is positive, shift-invariant and extends the usual limit, see the linked Wikipedia article. Of course, the terminology in various sources might differ.)
Your conditions can be rewritten as $$\liminf_{c\in D} f(c) \le \ell(f) \le \limsup_{c\in D} f(c).$$
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 $\mathcal U$ 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. We get multiplicativity from the fact that $\mathcal U$-limit is multiplicative. And the fact that $\mathcal U$ contains the tail filter helps with the condition about limit inferior and limit superior.)
The same construction is mentioned in the answer to: What is a generalized limit?
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? and Basic facts about ultrafilters and convergence of a sequence along an ultrafilter.
Sine you are interested in multiplicative functionals, this might be of interest, too: Every multiplicative linear functional on $\ell^{\infty}$ is the limit along an ultrafilter.