I'm trying to find the rate (or a bound for it) with which an iterated integral of the type

$$\int_{-h}^0 \int_{-h}^{t} A_s d B_s A_t d B_t$$

converges to zero (in probability/distribution) for $h \to 0$. Here, $B$ is a two-sided Brownian motion and $A$ is a bounded process with $A_0 = 0$. In my specific case, I already know that the rate has to be faster than $h^3$.

In the case of a single integral, there's something like the fundamental theorem of calculus (see Isaacson, https://www.jstor.org/stable/2239551) which says (in a more general form than stated here) that under certain conditions $$ \lim_{h \to 0} \frac{1}{B_{t+h} - B_t} \int_t^{t+h} A_s d B_s = A_t$$, thus implicitly giving the rate as $\sqrt{h \log(1/h)}$ by Levy's modulus of continuity for the Brownian motion. Is a result like this known for iterated integrals?

Even if not, I'd be thankful for possible (even vague) ways of bounding a (multiple) stochastic integral in terms of the integration interval.