Pursuant to helpful comments by user254433, I have decided to take another swing at this problem while reformulating it with a simplified example.
(Reformulated) General Problem: Generally speaking, I am trying to evaluate a definite integral of the form $\mathcal{I}(L)\equiv \int_0^{L}dy f(y,L) $ in the limit that $L\to 0$ (for $f$ real). Now, if the integrand of interest—and its derivatives in each variable—were finite at the "origin", $|f(0,0)|<\infty$, then I would simply attempt to Taylor expand the integral $\mathcal{I}(L)$ around $L=0$ and apply the "Leibniz integral rule" as follows,
\begin{align} \mathcal{I}(L) =\sum_{n=0}^\infty \left. \frac{d^n}{dL^n} \mathcal{I}(L) \right|_{L=0} \frac{L^n}{n!}&\equiv \sum_{n=0}^\infty \left[ \frac{d^n}{dL^n} \int_0^L f(y,L)dy \right]_{L=0} \frac{L^n}{n!}\\ &= f(L,L)|_{L=0}L+\left[f^{(0,1)}(L,L)+\frac{1}{2}f^{(1,0)}(L,L)\right]_{L=0}L^2+\mathcal{O}(L^3). \tag{1} \label{1} \end{align}
However, a naive application of this procedure to my integrand of interest is not possible because it diverges at the origin, $|f(0,0)|=\infty$. Presumably, I should be applying a different series expansion (is there some Laurent-type expansion for real functions, perhaps?), but I'm not sure which one or how to use it. [Maybe I need an "asymptotic expansion"? I really don't know what I should be doing here.]
Explicit (Simplified) Example: Consider the following (highly simplified) example integrand which diverges at the origin: $f(y,L)=y/L^2$. Now, for this example, it would be trivial to directly integrate $f$ over $y:[0,L]$ to obtain $\mathcal{I}(L)$=1/2, which is, moreover, independent of $L$, rendering the $L\to 0$ limit irrelevant. However, my actual integrand of interest involves a product of Bessel functions which I cannot directly integrate for finite, nonzero $L$.
Question: Ignoring the fact that this toy integrand can be directly integrated, how would one go about expanding $\mathcal{I}(L)$ around $L=0$ in this case? [Or if this toy integrand proves to be too trivial to illustrate the procedure, how would one go about expanding an integrand of your choice which also diverges in some way that would make a naive application of Taylor unworkable?]