How many low-order bits of an RSA modulus can we safely fix to 1?
Consider the similar RSA key generation process:
Select a random odd value $z$ between 1 and $2^\ell - 1$; choose $p$ as usual, fix the lower $\ell$ bits of $q$ to $z p^{-1} \bmod 2^\ell$ and chose $q$ with this constraint.
Now, this generates RSA keys with the same asymptotic [1] distribution as the standard RSA key generation process.
Now, the only difference between this procedure and yours is that you use a fixed $z = 2^\ell -1$, rather than a random one.  However, the attacker can determine the value $z$ used, and so us telling him which $z$ we picked doesn't give him any additional information.
So, the only question is &quot;does a specific value of $z$ yield a weakness?&quot;.  I do not know of any factoring method that is applicable to a specific $z$ value.

[1]: I say asymptotic, as the demonstration assumes that the lower $\ell$ bits of primes are evenly distributed over odd values; this has been proven to be true asymptopicly (for fixed $\ell$), it is a reasonable (but unproven) assumption for smaller primes.