= Solution
A <credibility estimate> combines a risk's own observed experience with a collective or prior mean, in the form $Z\widehat\mu_{\mathrm{experience}}+(1-Z)\mu_{\mathrm{collective}}$. The <credibility factor> $Z\in[0,1]$ is the weight given to that risk's experience. <Bayesian credibility> starts with a prior distribution for a latent risk parameter, updates it using observations, and estimates the conditional risk mean by its <Bayesian posterior> expected value under <squared-error loss>. In conjugate models this posterior mean can have exactly the displayed credibility form.
For the unequal policy exposures, define
$$
M=\sum_{i=1}^k m_i,\qquad s=\sum_{i=1}^k x_i.
$$
Conditional independence of the <binomial distribution> observations gives the likelihood, up to a factor not depending on $\theta$,
$$
L(\theta)\propto\prod_{i=1}^k\theta^{x_i}(1-\theta)^{m_i-x_i}
=\theta^s(1-\theta)^{M-s}.
$$
Multiplication by the <beta distribution> prior therefore gives <Beta-binomial conjugacy>:
$$
\boxed{\theta\mid x_1,\ldots,x_k\sim
\operatorname{Beta}(\alpha+s,\beta+M-s).}
$$
Both posterior parameters are positive, since $0\le s\le M$ and the prior parameters are positive.
The conditional target is $\mu(\theta)=m_{k+1}\theta$. To verify the quadratic-loss rule directly, for any proposed estimate $d$ and observed data $\mathcal D$,
$$
\mathbb E[(d-\mu(\theta))^2\mid\mathcal D]
=(d-\mathbb E[\mu(\theta)\mid\mathcal D])^2
+\operatorname{Var}(\mu(\theta)\mid\mathcal D).
$$
The variance term does not depend on $d$, so the unique <Bayes estimator under squared error loss> is the posterior mean. Applying the beta mean formula gives
$$
\boxed{\widehat\mu_{k+1}
=m_{k+1}\frac{\alpha+s}{\alpha+\beta+M}.}
$$
It is also the posterior expected future claim count, by the <tower property of conditional expectation>.
Let $p_0=\alpha/(\alpha+\beta)$ be the collective prior claim probability and $\widehat p=s/M$ the observed claims per policy. Then
$$
\frac{\alpha+s}{\alpha+\beta+M}
=\frac{M}{M+\alpha+\beta}\frac{s}{M}
+\frac{\alpha+\beta}{M+\alpha+\beta}\frac{\alpha}{\alpha+\beta}.
$$
Therefore the <exact beta-binomial credibility with unequal exposures> is
$$
\boxed{\widehat\mu_{k+1}
=Z\left(m_{k+1}\frac{s}{M}\right)
+(1-Z)\left(m_{k+1}\frac{\alpha}{\alpha+\beta}\right),
\qquad Z=\frac{M}{M+\alpha+\beta}.}
$$
The individual experience must be weighted by exposure: $s/M$ is the policy-weighted frequency, not the unweighted average of $x_i/m_i$ when the $m_i$ differ. The factor multiplies next-year expected counts after scaling both component probabilities by $m_{k+1}$.
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