= Solution
An exact finite <mixture model> follows from the <binomial theorem>:
$$
(2+\theta)^{y_1}\theta^a(1-\theta)^b
=\sum_{z=0}^{y_1}\binom{y_1}{z}2^{y_1-z}\theta^{a+z}(1-\theta)^b,
\qquad a=y_4,\quad b=y_2+y_3.
$$
Define positive <mixture weights>
$$
w_z=\frac{\binom{y_1}{z}2^{y_1-z}B(a+z+1,b+1)}{
\sum_{j=0}^{y_1}\binom{y_1}{j}2^{y_1-j}B(a+j+1,b+1)}.
$$
Here $B$ is the <Beta function>, so the component <probability distributions> are $\operatorname{Beta}(a+z+1,b+1)$. All coefficients are computable from the observed integer counts.
Use <inverse transform sampling> with the supplied $U$ to choose $Z$: writing $C_{-1}=0$ and $C_z=\sum_{j=0}^zw_j$, take $C_{Z-1}\leq U<C_Z$. Then recycle its position inside that interval,
$$
V=\frac{U-C_{Z-1}}{w_Z}.
$$
<Recycling a uniform random variable after discrete sampling> makes $V$ uniform independently of $Z$, since $\Pr(Z=z,V\leq v)=w_zv$. It is also independent of all supplied <random variables> with <gamma distributions>. Transform these by the <probability integral transform>:
$$
W_i=1-e^{-G_i},\qquad 1\leq i\leq N.
$$
They are independent <random variables> with <uniform distributions> because $G_i$ has a unit-rate <exponential distribution>.
Conditional on $Z=z$, put $k=a+z+b+1$ and $r=a+z+1$. Form the $k$ values $V,W_1,\ldots,W_{k-1}$ and return their $r$th <order statistic>. This uses at most the supplied $N$ <random variables> with <gamma distributions>, since $k-1=a+b+z\leq N$. The <uniform order statistic> formula gives
$$
\boxed{\Theta=\operatorname{order}_r(V,W_1,\ldots,W_{k-1}),\qquad
\Theta\mid Z=z\sim\operatorname{Beta}(a+z+1,b+1).}
$$
Averaging these component <probability density functions> with weights $w_z$ recovers exactly the normalized <posterior density>. Unused inputs can be discarded. When $N=0$, $Z=0$, $k=r=1$, and the output is $U$, as required. Interval endpoints and ties have <probability> zero; any consistent convention there is harmless. Exactness here concerns the mathematical algorithm; finite-precision arithmetic can be stabilized with logarithmic <mixture weights>.
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