My answer follows the suggestion to express the noisy outcome as a likelihood. I have changed the notation a bit (from the question) to handle some additional complications.
Let
\begin{equation}
p(y_t|\theta) = \textsf{Bernoulli}(\theta) = \begin{cases}
\theta & y_t = 1 \\
1-\theta & y_t = 0
\end{cases} ,
\end{equation}
where
\begin{equation}
p(\theta) = \textsf{Beta}(\theta|a,b) .
\end{equation}
Suppose we observe $y_1$. Then, as indicated in the setup to the question,
\begin{equation}
p(\theta|y_1) = \textsf{Beta}(\theta|a+y_1,b+1-y_1)
\end{equation}
and
\begin{equation}
p(y_2|y_1) = \textsf{Bernoulli}\left(y_2\Big|\frac{a+y_1}{a+b+1}\right) .
\end{equation}
Now suppose we don't observe $y_t$ directly. Instead we observe a noisy report, $z_t$, where
\begin{equation}
p(z_t|y_t) = \textsf{Bernoulli}(z_t|q_{y_t}) = \begin{cases}
q_{y_t} & z_t = 1 \\
1-q_{y_t} & z_t = 0
\end{cases} .
\end{equation}
In the question (as I understand it), $q_0 = .7$ and $q_1 = 1$.
We now have a complete model, which is to say we have the following joint distribution:
\begin{equation}
p(y_t,z_t,\theta) = p(z_t|y_t)\,p(y_t|\theta)\,p(\theta) ,
\end{equation}
where $(a,b,q_0,q_1)$ are known. Given this model, we have
\begin{equation}
p(y_t,\theta|z_t) = \frac{p(y_t,z_t,\theta)}{p(z_t)} ,
\end{equation}
where
\begin{equation}
p(z_t) = \sum_{y_t\in\{0,1\}} p(z_t|y_t)\,\int p(y_t|\theta)\,p(\theta)\,d\theta = \textsf{Bernoulli}\left(z_t\Big|\frac{a\,q_1+b\,q_0}{a+b}\right) .
\end{equation}
There are a number of ways one can proceed at this point to estimate the model (i.e., the compute the posterior distribution).
In addition there are a number of distributions we can address. Suppose we observe $z_1$. In addition to $p(\theta|z_1)$, we have
\begin{equation}
p(y_1|z_1) \qquad\text{and}\qquad p(y_t|z_1) ,
\end{equation}
where $t \ge 2$. The first of these two distributions is specific to $y_1$ since it is based on its own signal, while the second is generic since is applies to any $y_t$ for which we as yet have no signal. For $t \ge 2$, note
\begin{equation}
p(y_t|z_1) = \int p(y_t|\theta)\,p(\theta|z_1)\,d\theta .
\end{equation}
It is straightforward to extend this approach to allow for multiple observations where $y_{1:T} = (y_1, \ldots, y_T)$, $z_{1:T} = (z_1,\ldots,z_T)$, and
\begin{equation}
p(y_{1:T},z_{1:T},\theta) = p(z_{1:T}|y_{1:T})\,p(y_{1:T}|\theta)\,p(\theta) = \left(\prod_{t=1}^T p(z_t|y_t)\,p(y_t|\theta)\right) p(\theta) .
\end{equation}
We can compute a number of posterior distributions including $p(\theta|z_{1:T})$ as well as the specific distributions $p(y_t|z_{1:T})$ for $1 \le t \le T$ and the generic distribution
\begin{equation}
p(y_{T+1}|z_{1:T}) = \int p(y_{T+1}|\theta)\,p(\theta|z_{1:T})\,d\theta .
\end{equation}
