Bounding norms of expressions involving hadamard products

Question

Given square matrices $A, B, C$ and a vector $x$, I want to find an upper bound for expressions of the form $$ \| (A \circ B) C x\|_{2}. $$ Ideally, I want a bound that 'incorporates' the effect of $A$ on $C$ and $x$. What I mean by this is that the straight forward thing to do is $$ \| (A \circ B) C x\|_{2} \le \| A \circ B\|_{\text{op}} \|C\|_{\text{op}} \| x\|_{2} $$ and bound each term individually, but this approach removes any effect of $A$ on $C$ and $x$. I am wondering if there is a smarter approach that avoids doing this. Of course $(A \circ B)Cx \neq A \circ (B C)x $ etc, but are there any other distributive equalities/inequalities for the hadamard product that might work here?