A random variable is a counting variable does not only mean that it has natural number values. So, the number of sunny days in a year is not a counting random variable, because it is not the result of a counting process. Probably, a day is declared sunny if some burocratic criteria are fulfilled, like at least 5 hours clear sun, or whatever. It is not a count of independent events. Examples of count data is: number of auto accidents in new york, per day. Number of stillbirths in Guatemala, per day. These are counting independent events, which could as a first approximation be modelled via Poisson distribution or a poisson point process. I can see no such Poisson model lurking behind number of sunny days! For instance, have a look at my answer here: Goodness of fit and which model to choose linear regression or Poisson The arguments used there are irrelevant in your case.
Back to your question, if "the count is big". It is not the bigness in itself that matters, big counts could still be Poisson (but big counts would in practice often be clustered and some more complicated model than poisson would be needed). For number of sunny days in a year, you could sure try ordinary linear regression, as a starting point.
To elaborate on why "number of sunny days" is not a count variable. First, number of hours of (sufficiently strong) sunshine is measured at meteorological stations with a Campbell–Stokes recorder, see https://en.wikipedia.org/wiki/Campbell%E2%80%93Stokes_recorder They look like this:
and works by focusing the sun on a paper clip, and burning a path there when the sun is strong enough. Then one have to measure the length of the burnt path. That gives a measured variable, not a count variable! The underlying process is measurement, not counting. Then this is converted into a binary sunny/not sunny indicator by some arbitrary ("burocratic") cutoff. Hope this is a better explanation of my answer!