Given matrices $A$ and $B$, one can find an upper bound on the error $\epsilon = ||e^{A + B} - e^{A} e^{B}||$ in terms of the magnitude of the commutator between $A$ and $B$, $|| [A, B] ||$, as is done is papers such as https://arxiv.org/abs/1912.08854. Are there any analogous results which bound $\epsilon$ in terms of the commutator magnitude $|| [e^{A}, e^{B}]||$? (in particular, when $A$ and $B$ are Hermitian)?
I would suspect (naively) that some result in this direction may hold, seeing as when $A$ and $B$ are Hermitian, $[A, B] = 0 \Leftrightarrow [e^{A}, e^{B}] = 0$, but I have yet to find anything.
Without the Hermitian condition, this is not possible.
Choose $$A_0=\begin{pmatrix} 0&1&0&0\\0&0&1&0\\0&0&0&0\\0&0&0&0\end{pmatrix},\quad B_0=\begin{pmatrix} 0&0&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0\end{pmatrix};$$ Then for scalars $s,t$, one has $$W:=e^{tA_0}e^{sB_0}-e^{tA_0+sB_0}=\begin{pmatrix} 0&0&0& \frac13t^2s\\0&0&0&\frac12ts\\0&0&0&0\\0&0&0&0\end{pmatrix}.$$
Hence choosing $s=1/t$ and letting $t$ tend to infinity, $[tA_0,sB_0]$ is constant while $W$ tends to infinity. Or choosing $s=t^{-3/2}$, $[tA_0,sB_0]$ tends to zero while $W$ still tends to infinity.
Thus $\|e^{A}e^{B}-e^{A+B}\|$ can't be bounded in terms of $[A,B]$ only.
I'd suggest to also choose $A_0,B_0$ somewhat random Hermitian and estimate $\|e^{tA}e^{t^{-1}B}-e^{tA+t^{-1}B}\|$ to figure out if there are also Hermitian counterexamples.