Matrix Multiplication Rules Unfortunately I cannot post up the full question, but I need this little idea (whether its true or false to answer another question)

Let's say we have $v, w \in \mathbb{R}^n$ and $A,B$ be matrices with $B$ being symmetric. Then if

$v^tA^tBw = v^tBAw$

Is there a rule that can help me argue that $A^tB = BA$?  $(*)$

Somehow if I can show this then this means,

$A^tB^t = BA$ iff $(BA)^t = BA$. Hence $BA$ is symmetric.

[Side note: I have used $v^t$ to denote the transpose]. Again sorry I cannot give more context I cannot post up the question due to reasons. So my main point is $(*)$
Since you're trying to show $A^TB^T = BA \iff (BA)^T = BA$: I think you would be able to get by with the general statement
$$
(AB)^T = B^TA^T
$$
which holds for any matrices $A,B$.  Here's a proof.
I'm not quite sure on your wording, but assuming you mean $v^tA^tBw = v^tBAw$ holds for all $v,w\in\mathbb{R}^n$ and you want to prove that $A^tB=BA$, then you can write $v^tA^tBw = v^tBAw$ as $v^tC=v^tD$, where $C=A^tBw,D=BAw$. Thus we see that $C$ has to equal $D$. Now repeat this process.