I'm not an expert on integral representations, but I don't think such a reference should exist. The method of automorphic representations are crucial to Jacquet's approach which, analogous to Tate's thesis, is to define the local Rankin-Selberg L-factor as a gcd of zeta integrals $Z(s,\phi_v \times \phi_v')$ where $\phi_v, \phi_v'$ run over vectors in the associated local representations. (There are also parameters for additive characters and Eisenstein series, which I supress.) It seems to me there should be no classical translation of this approach, as there is no classical version of varying $\phi_v, \phi_v'$ locally in a $GL_2(\mathbb Q_v)$-representation.

So now you may ask: how can you relate the $L$-function to a classical Rankin-Selberg integral?

In the unramified case, Jacquet proves that $Z(s,\phi_v \times \phi_v') = L(s, \phi_v \times \phi_v')$ where $\phi_v, \phi_v'$ are new vectors. This means the classical approach to Rankin-Selberg should give the right factors at the unramified primes. However, Jacquet does not determine "test vectors" $\phi_{v}, \phi_{v}'$ where the zeta integral spits out the $L$-factor on the nose in general. Indeed, determining test vectors is a nontrivial problem and they are known not to always exist in more general settings (i.e., the gcd may not be attained for any choice of $\phi_{v}, \phi_v'$). In this setting however, test vectors probably do exist, and as I recall under some conditions they were determined in the (unpublished but accessible) thesis of Kim Mi-Kyung, a fairly recent student of Cogdell.)

The problem is that without knowing how $Z(s,\phi_v \times \phi_v')$ relates to $L(s, \phi_v' \times \phi_v')$ when $\phi_v, \phi_v'$ are newvectors for ramified representations, you can't relate the complete $L$-function to a classical Rankin-Selberg integral exactly. (One also needs the analogous archimedean theory; in this case Jacquet does determine test vectors.) Possibly one can do more with the classical Rankin-Selberg integrals now in light of Kim's thesis (I don't remember exactly what she did), but I'm not aware of any such work so far.

If you still want to understand the idea of Jacquet's approach, besides Bump's book and his surveys (discussing more general Rankin-Selberg cases), Cogdell has several nice notes on the Rankin-Selberg method which include the more general setting of $GL(n) \times GL(m)$ (e.g., his Fields lecture notes).