Is a t-test applicable here in comparing two proportions? I have 3 categories of tissues examined by microscopy (A , B and C, 12 of each). I want to test whether the proportion of tissues staining by a particular dye, differs in the three categories. Each  of the 36 tissue specimens is stained and examined under the microscope. The data is collected in the form of proportion of cells staining positive for the dye in each tissue specimen.
Now, I understand that the proportions have a binomial distribution and t-test is probably not appropriate because the variance of proportions is known. But I don't know how to apply the z-test to detect the difference in proportion here, since the data collected itself is in the form of "proportion" for each specimen. Do I calculate the mean proportion ($p_A,p_B,p_C$ for each category) by averaging the recorded proportions under each observation and then apply the z-test?
As an alternative thought, am I not estimating the proportion from my study population by averaging the proportions themselves. Thus, since I am actually estimating the true population proportion from sample itself, would the assumptions of z-test hold here?

For completeness, let's first examine each test and their assumptions and understand if using a t test in place of the z test is OK.
The reason we do a t test at all is because we have to estimate the standard error of the data, which requires an estimate of the variance.  Were we to do the experiment again, that estimate might change.  This is known as sampling variability and that additional variability induced by estimating the variance makes the tails of the sampling distribution slightly fatter than a normal.  So in short, when we need to estimate the variance we use a t test.
Note that we don't need to estimate the variance for a binomial outcome, its a function of the estimated mean.  Recall that if $X$ is binomial random variable, then
$$ \operatorname{Var}(X) = np(1-p) $$
so if we have an estimate of $p$, we also have an estimate of the variance.  This is not strictly true with non-binomial data.
Is there harm in using a t test in place of a z test?
My intuition says that there should be.  This is because the critical value for the t-test -- especially when you have 12 replicates per condition -- will be a lot larger and hence should hurt power.  But, if you simulate the test across a range of baseline probabilities and effect sizes (shown below) we see that the choice of test doesn't really impact power
Shown below, I've run each test 1000 times for each indicated value of the true outcome (p) and an effect size (deltap).  In these simulations, it doesn't seem to matter much which you choose, so if you are going to choose a test I would pick the z test to be consistent with the literature and other statistical approaches.

