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 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.)

EDIT: The OP mentioned in comments that the definition of *Banach limit* given in the question (i.e., with directed sets) can be found in Schechter's Handbook of Analysis and Its Foundations, see [page 318](https://books.google.com/books?hl=en&id=eqUv3Bcd56EC&pg=PA318) and also in Howard, Rubin: Consequences of the Axiom of Choice, see [page 63](https://books.google.com/books?id=ffXxBwAAQBAJ&pg=PA63), Form 372. (If we work on arbitrary directed sets, we do not have a natural way to talk about shift-invariance. So this is not a generalization of the notion of Banach limit on $\mathbb N$ mentioned in the first paragraph.)
 
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Your conditions can be rewritten as<sup>1</sup>
$$\liminf_{c\in D} f(c) \le \ell(f) \le \limsup_{c\in D} f(c).$$

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](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.<sup>2</sup> 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?](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).

Since 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.](https://math.stackexchange.com/q/175553)

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<sup>1</sup>For more on limit superior/inferior of a net, see: [Limsups of nets](https://math.stackexchange.com/q/188722) and [About the notion of limsup and liminf](https://math.stackexchange.com/q/49699)

<sup>2</sup>Limit along an ultrafilter with values in a compact space always exists. The proof is given, for example, in this answer: [Basic facts about ultrafilters and convergence of a sequence along an ultrafilter](https://math.stackexchange.com/q/51476#51483).