I am aware of how one can represent the Cartesian product of two sets, say $A$ and $B$. However, is there are standard way to represent the scalar product of a value and a set/multiset? As a simple example (with a multiset), let $P = \{1, 2, 3, 4, 2\}$ and $q = 2$. Then $$q \cdot P := \{(1 \cdot 2),(2 \cdot 2),(3 \cdot 2), (4 \cdot 2), (2 \cdot 2)\} = \{2, 4, 6, 8, 4 \}. $$
Could this be the appropriate notation for a scalar product? I'm not entirely certain scalar multiplication of a value and a set exists as I haven't been able to find it anywhere in books or online––if this is the case, is it because set are immutable? In case it is asked, I am unfortunately unable in this example to make $P$ a vector and do the same operation.
The notation you suggest is very common notation in contexts where scalar multiplication makes sense. For example, in the study of fractal geometry, fractal sets often display self-similarity. It is not uncommon to see a self-similar subset of $\mathbb{R}^n$ described as a set $F$ having the property that $$ F = \bigcup_{j=1}^{N} c_j F + b_j, $$ for some collection of $c_j \in \mathbb{R}$ and $b_j\in\mathbb{R}^N$. Here, each $c_j$ scales the set $F$, then $b_j$ translates the scaled copy. Both the scalar multiplication and the addition make perfect sense, and the notation is exactly what you suggest.
Slightly more generally, let $V$ be a vector space over a field $k$ (for example, $V = \mathbb{R}^n$ is a vector space over $k = \mathbb{R}$), let $P \subseteq V$, and let $q \in k$. Then $$ q\cdot P = qP = \{ qp : p \in P \}. $$ Even more generally, if $P$ is any set with a structure which supports some kind of multiplication, and $q$ is any object such that $qp$ makes sense for $p\in P$, then the notation is likely to be understood.
As a basic example, you might encounter the notation $$ \mathbb{Z} / p\mathbb{Z} \qquad\text{(where $p$ is prime)} $$ in a typical undergraduate course in abstract algebra. This is the quotient of $\mathbb{Z}$ (the integers) with $p\mathbb{Z}$ (multiples of $p$, i.e. the set $\{np : n\in\mathbb{Z} \}$).
As an addendum, there are also notions of sums and differences of sets with this kind of notation. If $A$ and $B$ are two subsets of some space in which addition and subtraction are defined, then we can define the Minkowski sum and difference of $A$ and $B$ as $$ A\pm B := \{ a\pm b : a\in A \land b\in B \}. $$ A similar notation could easily be adopted for a "Minkowski product" or "Minkowski quotient" (though I would be careful with those terms, as they may have other meanings). Indeed, I recently came across a paper which uses the notation $$ CD := \{ cd : c\in C \land d\in D \}, $$ where $C, D \subseteq \mathbb{R}$. That same paper also uses the notation $$ 1/\mathbb{N} := \{ \tfrac{1}{n} : n\in\mathbb{N} \}, $$ which is consistent with the multiplicative notation.