In the Quadratic Sieve, why restrict the factor base?
The condition $\genfrac(){}{}{a}{b}$=+1 doen't matter as the product of 2 QNR is a QR.In fact the idea of locking for numbers satisfying $x^2-y^2=k.n$ was firstly introduced by Gauss (1801) and was developped by Kraichick (1920) by searching for special sequences of the form $Q(x_i)=x_i^2 -n$ which factorize over small primes $\prod Q(x_i)=v^2$. In the exemple you gave the number of QNR are even, then in the product only QR factors are present.
Selecting larger primes outside the factor base, need additional ressource in storage and computation and doen't give additionnal advantage in the form of Fermat difference.