Let me first present the problem in its most general form, and then work down to a specific case of interest and my motivation.
Suppose $f_1: \mathbb{R}^2 \rightarrow \mathbb{R}_{\ge 0}$ and $g: \mathbb{R} \rightarrow \mathbb{R}_{\ge 0}$ are known functions, but $f_2: \mathbb{R}^2 \rightarrow \mathbb{R}_{\ge 0}$ is unknown. How would one solve the integral equation \begin{align*} \int_{\mathbb{R}}f_1(x, y) f_2(x, y) dx = g(y) \end{align*}
More specifically, I'm interested in the probabilistic interpretation when we let $f_1(x, y) = \mathbb{P}(A|x, y)$, $f_2(x, y) = f(x|y)$, and $g(y) = \mathbb{P}(A|y)$, where $A$ is some event. The problem boils down to this: given we know the conditional probability of $A|x,y$, find a (not "the", for it may not be unique) conditional distribution for $x|y$ such that $A|y$ is some desired functional form. To be even more specific, I'm interested when \begin{align*} f_1(x, y) &= \frac{e^{x + \alpha_0 + \alpha_1 y}}{1+e^{x + \alpha_0 + \alpha_1 y}} \\ g(y) &= \frac{e^{\beta_0 + \beta_1 y}}{1+e^{\beta_0 + \beta_1 y}} \end{align*} where $\beta_0, \beta_1$ are known, and $\alpha_0, \alpha_1$ are adjusted to reach the final $g(y)$.
Any help would be appreciated!