The Poland--Scheraga model, introduced in the 1970's, is a reference model to describe the denaturation transition of DNA. More recently, it has been generalized in order to allow for asymmetry in the strands lengths and in the formation of loops: the mathematical representation is based on a bivariate renewal process, that describes the pairs of bases that bond together. In this paper, we consider a disordered version of the model, in which the two strands interact via a potential $βV(\hatω_i,\barω_j)+h$ when the $i$-th monomer of the first strand and the $j$-th monomer of the second strand meet. Here, $h\in\mathbb R$ is a homogeneous pinning parameter, $(\hatω_i)_{i\geq 1}$ and $(\barω_j)_{j\geq 1}$ are two sequences of i.i.d.~random variables attached to each DNA strand, $V(\cdot,\cdot)$ is an interaction function and $β>0$ is the disorder intensity. Our main result finds some condition on the underlying bivariate renewal so that, if one takes $β,h\downarrow0$ at some appropriate (explicit) rate as the length of the strands go to infinity, the partition function of the model admits a non-trivial, i.e. disordered, scaling limit. This is known as an \textit{intermediate disorder} regime and is linked to the question of disorder relevance for the denaturation transition. Interestingly and surprisingly, the rate at which one has to take $β\downarrow0$ depends on the interaction function $V(\cdot,\cdot)$ and on the distribution of $(\hatω_i)_{i\geq 1}$, $(\barω_j)_{j\geq 1}$. On the other hand, the intermediate disorder limit of the partition function, when it exists, is universal: it is expressed as a chaos expansion of iterated integrals against a Gaussian process~$\mathcal{M}$, which arises as the scaling limit of the field $(e^{βV(\hatω_i,\barω_j)})_{i,j\geq 0}$ and exhibits strong correlations on lines and columns.