Sifting for small split primes of an imaginary quadratic field in a given ideal class

Let $D \gt 3$, $D\equiv 3\pmod{4}$ be a prime, and let $\mathcal {C}$ be an ideal class in the field $\mathbf {Q}(\sqrt{-D})$. We give a new proof that $p(D,\mathcal {C})$, the smallest norm of a split prime $\mathfrak {p}\in \mathcal {C}$, satisfies $p(D,\mathcal {C})\ll D^L$ for some absolute constant $L$. Our proof is sieve-theoretic. In particular, this allows us to avoid the use of log-free zero-density estimates (for class group $L$-functions) and the repulsion properties of exceptional zeros, two crucial inputs to previous proofs of this result.