How do we define the density of a discrete random variable? Of a pair of mixed random variables? Suppose $X \sim \text{Binomial}(n, \theta)$ and $\theta \sim \text{Unif}(0, 1)$, so that
$$P(X = k \mid \theta) = {n \choose k} \theta^k (1 - \theta)^{n - k}.$$
In order to compute the posterior, I need to write down the joint "density" of $X$ and $\theta$.
With respect to what measure does $(X, \theta)$ have a density? I guess $X$ has a density with respect to counting measure (is this accurate?) and $\theta$ with respect to lebesgue measure. What about the pair?

As suggested in the comments, the canonical choice would be the Radon-Nikodym derivative with respect to the product of the Lebesgue and the counting measure.
Working on the probability space $\mathbb R^2$ equipped with this product directly, the Radon-Nikodym derivative is $(\theta,x)\mapsto\unicode{120793}\{\theta\in[0,1],x\in\mathbb Z\}\binom{n}{x}\theta^x(1-\theta)^{n-x}$. The way you defined the probability measure, I would read it as the semi-direct product, which does not require a Radon-Nikodym derivative. I'm also not entirely sure which distribution you refer to with the posterior, but if you mean the pushforward, using the kernel approach above you obtain it directly as discussed here. Put, simply, it's just $\mathbb P(X=k)=\mathbb E[\binom{n}{k}Y^k(1-Y)^{n-k}]$.
