How can I show that the Cook-Levin theorem does not relativize?

Question

The following is an exercise which I am stuck at ( source: Sanjeev Arora and Boaz Barak; its not homework ) :

Show that there is an oracle $A$ and a language $L \in NP^A$ such that $L$ is not polynomial-time reducible to 3SAT even when the machine computing the reduction is allowed access to $A$.

What I tried was, take $A$ to be the oracle to halting problem and let $L=\{1^n | \;\exists \; \langle M,w \rangle \; \text{s.t.} \; |\langle M,w \rangle|=n \; \text{ and Turing machine M halts on w} \} $.

With this assignment I ensure $L \in NP^{A}$ and $L$ is not polynomial reducible to 3SAT if oracle is not provided to the machine carrying out reduction. Although to map an instance $1^n$ I would have to search through $2^n$ strings even if oracle is provided to the reduction machine. But this does not seem like a proof for absence of polynomial reduction in this case.

Is there a way to prove it using the same example ? Is there a simpler example ?