The z-distribution (which is the same as the normal distribution) is always better than the t-distribution if and only if you have knowledge of the standard deviation of the population ( ).
Since you are talking about a binomial distribution that implies you have knowledge about . Therefore the normal distribution is a better approximation than the t-distribution.
There is no single distribution that always gives the best estimate for a confidence interval.
It depends on the (assumed) distribution of the population (normal, binomial, uniform, poisson, ...).
Btw, in the tails of any distribution the probability density becomes very unreliable in practice, since there are always other effects that are unaccounted for.
In practice extreme outcomes are more probable than any normal distribution predicts.