If $K$ is a number field, a result from Bach tells us that the primes in $K$ of norm smaller that $12 (\log |\mathrm{Discriminant}(K)|)^2$ generate the ideal class group $\mathrm{Cl}_K$. Is there any known bound if we require the generating primes to be unramified in $K/\mathbb Q$? (i.e. a result of the form "prime ideals in $K$ above the prime numbers unramified in $K/\mathbb Q$ and smaller than $f(K)$ generate the ideal class group")

In the particular case I study, $[K:\mathbb Q] = 4$. With algebraic methods I can find a bound polynomial in the class number $h_K$ and logarithmic in $|\mathrm{Discriminant}(K)|$, but I am sure we can get rid of the $h_K$, or at least change it for a $\log h_K$, but I'm afraid this would require analytic methods...