Filter before or after multiplication of two signals?
A personal rule: in general, it can be better to perform non-linear operations before linear ones. One reason behind that is that a lot of practical concerns are related to outliers or suspect behavior, which can  easily be smoothed out (and become indistinct from other signals) by  linear filters.

Let me reformulate. If $f_i$ denote filters, and $s_i$ signals, should one do $f_0 \ast(s_1 .s_2)$ or $(f_1 \ast s_1)(f_2 \ast s_2)$? Without a better knowledge on the spectra of the signal, the nature of the noise and the numerically objective, I have no definite answer. The second one is apparently more flexible, become you can play on two filters. 

If the result is some kind of average, I would filer the product, as  averaging is filtering, and it commutes with filtering. Thus, you have more chance to preserve the average after filtering the product, if the noise is low. Since two signals like $+1,-1,+1,-1\ldots$. Any even-length avering will turn them into $0$, while their product is flat with average 1.

I cannot see why you will have less numerical problems.
Noise is added "logarithmically", but the same happens to signals. But yes, modulation can be troublesome, but two cosine with zero mean will be multiplied into a non-zero mean signal, and this is what you want to measure.

In some cases though, to preserve contrast in image, homomorphic filtering convert a product into a sum to filter separately the two components (cf. "The second one is apparently more flexible"). Finally, you can use non-linear filters on the product as well, in case of complicated noises.  A related question on the ordre of operations can be found in Is convolution distributive over multiplication?