Consider the following integral $$ pv\int_0^{\infty}e^{N(-2Ax+A\log x)}\frac{e^{-B\log x}}{1-2x}dx $$ where $A,B>0$ and we take the Cauchy principal value at $x=1/2$. I am interested in obtaining the asymptotics when $N$ is very big. The first thing I thought of was some variant of Laplace's method but I am unsure if I can proceed here, because of the singularity at $x=1/2$. So, my question is, is some version of the Laplace's method applicable here to obtain the big $N$ asymptotics? and if so, how should I proceed?

You may estimate the integral on $[1,\infty)$ somehow (it does not affect the asymptotics), after that consider the integral of $\int_0^{1/2}f_N(1/2-t)+f_N(1/2+t)dt$, where $f_N$ is your function. This already does not have a singularity at 0 and you may apply Laplace method.

Mathematica tells us that the principal value can be evaluated in closed form thus:

$$e^{-A} 2^{-A N+B-1} \left((-1)^{A N-B+1} \Gamma (-B+A N+1) \Gamma (B-A N,-A)-i \pi \right).$$

(OK, the $i \pi$ is a little suspicious, but the appearance of incomplete gammas is clearly correct, once you replace terms like $\exp(C \log x)$ by $x^C.$).