Integer factorization still hard with Hamming weight hypothesis?
I suspect that this might be vulnerable to a combinatorial factoring attack.

In this attack, we look at possible solutions to $pq = n \bmod 2^k$, and then extend $p$ and $q$ one bit to list the possible solutions to $pq = n \bmod 2^{k+1}$

Now, if we have no further information about $p$ and $q$, this turns out to be no more efficient than brute force search (as at each step, the number of possibilities doubles, and we have no way to distinguish likely versus unlikely possibilities).

However, in your scenario, we know that both $p$ and $q$ are strongly biased toward 0 bits, and so it would make sense to order our search to explore the possibilities with $p$ and $q$ being mostly 0 first.  The immediately obvious approach would be to keep a list of possible solutions to $pq = n \bmod 2^k$, ordered by hamming weight of $p$ and $q$ (and $k$ somehow); and explore the long solutions with low hamming weight first.

It would seem plausible that this might quickly find a solution if the hamming weight was sufficiently small.  How well would it work in practice (and how small is 'small')?  I suspect that's something that would need to be experimented with.