It's a matter of careful interpretation.
Duality :
$$x[n] \iff X[k]$$
then
$$X[n] \iff Nx[-k]\quad , \tag{1}$$
where $X[k]$ is the N-point DFT of N-point $x[n]$.
Now, let $x_1[n]$ and $x_2[n]$ be sequences of length $N$, whose N-point DFTs are $X_1[k]$ and $X_2[k]$, respectively.
Given that :
$$\mathcal{DFT}(x_1[n] \circledast x_2[n]) = X_1[k]X_2[k]$$
then duality suggests:
$$ X_1[n] X_2[n] \iff N x_1[-k] \circledast x_2[-k] \tag{2}$$
, which is correct indeed.
Looking at eq.(2), however, the sequences on the left ($X_1, X_2$) are freq-domain sequences, and the sequences on the right ($x_1,x_2$) are the originating time-domain sequences.
But this is opposite of the usual DFT property notation where the sequences on the left are time-domain sequences, and those on the left are their forward DFT sequences which express the corresponding property.
So what's the forward DFT of $X_1[n]$...? It's $N x_1[-n]$ (from Eq.(1)), and also for $X_2[n]$ is $N x_2[-n]$.
Now re-interpret Eq.(2) as follows:
$$ X_1[n] X_2[n] \iff \frac{1}{N} (N x_1[-k]) \circledast (N x_2[-k]) \tag{3}$$
or further:
$$ X_1[n] X_2[n] \iff \frac{1}{N} \mathcal{DFT}\{ X_1[n] \} \circledast \mathcal{DFT}\{ X_2[n] \} \tag{4}$$
Now in this expression in Eq.4, the sequences on the left are arbitrary time-domain interpreted sequences, and the ones on the right will the their corresponding forward DFT sequences.
Finally, names of the sequences $X_1,X_2$ can be replaced with usual DFT notation, with $x[n] \iff X[k]$, and $h[n] \iff H[k]$ as DFT pairs:
$$ \mathcal{DFT}\{ x[n] h[n] \} \iff \frac{1}{N} X[k] \circledast H[k] \tag{5}$$
, which is the result you were looking for.
