Band limited discrete signals division
I've found an estimator that can work for a lot of similar situations, the idea is that you can rewrite the division:
$$ x(t) = z(t) \cdot y(t) \tag1$$
The problem now is to estimate $ z(t) $ that solves the equation.
One can start to find a constant coefficient that minimize the 2-norm. This coefficient will be the linear regression slope without intercept term ($\beta$), hence you can write an approximation of $ z(t) $ that minimize the 2-norm:
$$ x(t) \approx \beta \cdot y(t)\tag2 $$
This estimator is constant, hence it is band limited, and you can estimate the same $ \beta $ from the sampled data, in fact be valid that
$$ x[n] \approx \beta \cdot y[n]\tag3 $$
Obviously this estimation is quite useless but you can extend this approach in a kind of FIR way , first.
You could make a rectangular window and estimate the linear regression slope on $N$ elements.
$$ x[n] \approx \beta(x[(n-N),..,N], y[(n-N),..,n]) \cdot y[n] \tag4$$
I don't like a lot this estimator but should limit aliasing if $ N $ is  large enough.
I instead prefer an IIR like approach, using an exponential window. You can do this using a 0-order RLS algorithm. You can choose a bandwidth choosing the forgetting factor $ \lambda = e^{-\frac{1}{W_s}}$,
where $ W_s $ is the width in samples of the exponential window, that can be expressed in function of cut off pulsation ($\omega_c$) using the relation
$$ W_s = \frac{f_s}{\omega_c} \tag 5$$
where $f_s$ is the sampling frequency.
With RLS you can estimate the slope $\beta$ in a time-variant way, with a limited band in discrete domain, where the bandwidth is limited by $ \omega_c $. Note that in this case (0-order) RLS becomes a scalar problem so it is very easy to implement.