Upper bound for $(AB-BA)x$

Question

Given matrices $A,B\in\mathbb{R}^{n\times n}$where matrix $A$ is a diagonal matrix and $B$ is an upper triangular matrix. I'm looking for an upper bound for the expression \begin{align*} (AB-BA), \end{align*} in the form of i.e. \begin{align*} \lVert(AB-BA)\rVert_2\leq\lVert f(A)\rVert_2\cdot\lVert g(B)\rVert_2. \end{align*} My intent is to create an upper bound which seperates the effect of $A$ and $ B$. We assume that: \begin{align*} A\neq c\cdot I,\quad c\in\mathbb{R} \end{align*} But, ideally, the upper-bound should reflect it when this property is the case. For instance we can define a rough upper bound: \begin{align*} \lVert(AB-BA)\rVert_2\leq(2\lVert A\rVert_2\lVert B\rVert_2). \end{align*} But it doesn't reflect the scenario where $A=c\cdot I$ holds (i.e. AB-BA=0). Any suggestions would be greatly appreciated!