The beta-binomial pmf is
$$f_X(x) = {n \choose x}{B(x+\alpha, n-x+\beta)\over B(\alpha, \beta)}$$
where $B$ is the beta function.
The numerator is the issue here when trying to separate data from parameters. I tried using
$$B(x+\alpha, n-x+\beta) \propto \Gamma(y+\alpha)\Gamma(n-x+\beta)$$
and the fact that $x$ is discrete but that led me to a polynomial where parameters and data are still intertwined through the exponents and coefficients.
Is this solvable at all?
Here, we have $p(y|\theta)$ is our data from a Binomial distribution, parameters $n,p$. And $p(\theta)$ is from a Beta distribution, parameters $a=1,b=1$ for simplicity. The final image shows that the hyper parameters are just additive, where $dbeta$ is essentially the Binom-Beta distribution.